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Answer Key ALGEBRA 2 and TRIGONOMETRY AMSCO SCHOOL PUBLICATIONS, INC. 315 HUDSON STREET, NEW YORK, N.Y. 10013 AMSCO
Answer Key
ALGEBRA 2and TRIGONOMETRY
AMSCO SCHOOL PUBLICATIONS, INC.315 HUDSON STREET, NEW YORK, N.Y. 10013
A M S C O
14580AK_FM.pgs 3/26/09 12:07 PM Page i
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Copyright © 2009 by Amsco School Publications, Inc.
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Printed in the United States of America1 2 3 4 5 6 7 8 9 10 14 13 12 11 10 09
14580AK_FM.pgs 3/26/09 12:07 PM Page ii
Contents
Answer Keys
For Enrichment Activities 246
For Extended Tasks 255
For Suggested Test Items 261
For SAT Preparation Exercises 269
For Textbook Exercises
Chapter 1 271
Chapter 2 274
Chapter 3 277
Chapter 4 282
Chapter 5 291
Chapter 6 299
Chapter 7 303
Chapter 8 308
Chapter 9 312
Chapter 10 319
Chapter 11 324
Chapter 12 334
Chapter 13 343
Chapter 14 345
Chapter 15 349
Chapter 16 359
iv
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246
7.
8. (2x 1 y)(4x2 2 2xy 1 y2)9. (x 2 2y)(x2 1 2xy 1 4y2)
10. (5 2 3d)(25 1 15d 1 9d2)11. (4x 1 3y)(16x2 2 12xy 1 9y2)12.
13.
Enrichment Activity 2-5: InvestigatingRatios and Growth Rate in Leaves
Students answers will all differ as they havedifferent size leaves. If the length to width ratios arevery similar, students should conclude that the rate ofgrowth in their tree or bush is constant. If the lengthto width ratios vary a lot, they should conclude thatthe growth rate for their tree or bush is not constant.
Students should be assessed on the followingcharacteristics:
a. the accuracy of their measurementb. the construction of their data table and scatter
plotc. the accuracy of their computations with the
calculatord. their knowledge of ratio and averagee. the neatness of their workf. their ability to follow directionsg. how well they work with others if the activity is
done as a grouph. their ability to reach a conclusion
Enrichment Activity 3-2:A Square-Root Algorithm
1. 57 2. 72 3. 91 4. 395. 2.6 6. 4.1 7. 5.9 8. 9.4
5 (a 2 b)(a 1 b)(a2 1 b2)a4 2 b4 5 (a2 2 b2)(a2 1 b2)5 (a 2 b)(a 1 b)(a2 1 b2)5 (a 2 b) fa2(a 1 b) 1 b2(a 1 b)g5 (a 2 b)(a3 1 a2b 1 ab2 1 3)
1 b3(a 2 b)5 a3(a 2 b) 1 a2b(a 2 b) 1 ab2(a 2 b)
1 ab3 2 b45 a4 2 a3b 1 a3b 2 a2b2 1 a2b2 2 ab3
a4 2 b4
5 (a 2 b)(a2 1 ab 1 b2)5 a2(a 2 b) 1 ab(a 2 b) 1 b2(a 2 b)5 a3 2 a2b 1 a2b 2 ab2 1 ab2 2 b3
a3 2 b3Enrichment Activity 1-5:On the Ins and Outs
1. a. 110 2 108 5 2b. 380 2 378 5 2c.
d. (x 1 1)(x 1 2) 2 x(x 1 3)5 x2 1 3x 1 2 2 x2 2 3x5 2
2. The products differ by 2.3. a. 130 2 112 5 18
b. 598 2 580 5 18c. 11,128 2 11,110 5 18d. (x 1 3)(x 1 6) 2 x(x 1 9)
5 x2 1 9x 1 18 2 x2 2 9x5 18
4. The products differ by 18.5. The products differ by 32.6. a. (x 1 2)(x 1 4) 2 x(x 1 6)
5 x2 1 6x 1 8 2 x2 2 6x5 8
b. (x 1 5)(x 1 10) 2 x(x 1 15)5 x2 1 15x 1 50 2 x2 2 15x5 50
c.
d. (x 1 k)(x 1 2k) 2 x(x 1 3k)5 x2 1 3kx 1 2k2 2 x2 1 3kx5 2k2
7. If the numbers increase by any real number k,then the difference of the product is 2k2.
8. 9
Enrichment Activity 1-6: Factoring theSum and Difference of Two Cubes
1. (x 2 2)(x2 1 2x 1 4)2. (x 1 4)(x2 2 4x 1 16)3. (x 2 4)(x2 1 4x 1 16)4. (x 1 5)(x2 2 5x 1 25)5. (x 2 2y)(x2 1 2xy 1 4y2)6.
5 (a 1 b)(a2 2 ab 1 b2)5 a2(a 1 b) 2 ab(a 1 b) 1 b2(a 1 b)5 a3 1 a2b 2 a2b 2 ab2 1 ab2 1 b3
a3 1 b3
5 725 2(62)5 x2 1 3(6x) 1 2(62) 2 x2 2 3(6x)
(x 1 6)(x 1 2(6)) 2 x(x 1 3(6))
5 25 302 1 3(30) 1 2 2 302 2 3(30)
(30 1 1)(30 1 2) 2 f30(30 1 3)g
Answers for Enrichment ExercisesAnswers for Enrichment Exercises
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Enrichment Activity 3-5:A Radical Sequence
1.
2.
3.
4. The sequence has a common ratio, r.
5.
6.
7.
8.
9.
10. 1.618, the golden ratio
Enrichment Activity 4-5:The Method of Finite Differences
1. f(x) 5 x2 2 3x 1 82. f(x) 5 2x2 1 5x 2 33. f(x) 5 3x2 2 4x 2 154. f(x) 5 x3 2 2x2 1 5x 2 3
Enrichment Activity 4-7:The Difference Quotient
1. a. 2x 1 h b. 2x2. a. x3 1 3hx2 1 3h2x 1 h3
b. 3x2 1 3hx 1 h2
c. 3x2
3. a. x4 1 4hx3 1 6h2x2 1 4h3x 1 h4
b. 4x3 1 6hx2 1 4h2x 1 h3
c. 4x3
4. a. x5 1 5hx4 1 10h2x3 1 10h3x2 1 5h4x 1 h5
b. 5x4 1 10hx3 1 10h2x2 1 5h3x 1 h4
c. 5x4
5. a.
b. Possible answer: The value of the differencequotient when h 5 0 for f(x) 5 xn is nxn21.
6. a. 6x5 b. 9x8 c. nxn21
5x44x33x22x
511 1 5!5
2Q7 1 3!52 RQ1 1 !5
2 R 5 74 1
10!54 1 15
4
A2 1 !5B Q1 1 !52 R 5 1 1
3!52 1 5
2 57 1 3!5
2
58 1 4!5
4 5 2 1 !5
Q3 1 !52 RQ1 1 !5
2 R 53 1 4!5 1 5
4
53 1 !5
2Q1 1 !52 R2
51 1 2!5 1 5
2 56 1 2!5
2
1 1 !52
7 1 3!52 , 11 1 5!5
2
4 1 2!52 5 2 1 !5
1 1 !52 1 1 5
1 1 !5 1 22 5
3 1 !52
247
Enrichment Activity 5-6A:Complex Number Operations,Vectors,and Transformations
1. a.
B 5 23 1 5i C 5 25 2 3iD 5 3 2 5i E 5 5 1 3i 5 A
b. F 5 23 1 5i 5 Bc. Answers will vary: multiplication by i is
equivalent to a counterclockwise rotation of90° about the origin. Multiplication by i2 (or21) is equivalent to a rotation of 180°.Multiplication by i3 (or 2i) is equivalent to acounterclockwise rotation of 270° about theorigin. Multiplication by i4 (or 1) is theidentity transformation.
d. Answers will vary:Point symmetry in the originRotational symmetry of 90° (as well as 180°and 270°) about the origin
Line symmetry through , through 2. a–c. yi
xO
A
B
CD
E
F
G
H30
10
20
10210
BOCg
AOCg
yi
xO
A, E
B
C
D
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b. B 5 4 2 2i C 5 2 2 6iD 5 24 2 8i E 5 212 2 4iF 5 216 1 8i G 5 28 1 24iH 5 16 1 32i
3. a–b.
B 5 2i C 5 22 1 2i D 5 24E 5 24 2 4i F 5 28i G 5 8 2 8iH 5 16 I 5 16 1 16i
c. (1) H, P, X(2) D, L, T(3) B, F, J, N, R, V, Z
4. Case 1: Multiplication by 0 1 0i is not atransformation of the plane because every pointmaps to 0, a single point.
Case 2: If a � 0 but b 5 0, then multiplication bya, is a dilation of a. A dilation of a is a specialcase of spiral similarity where no rotation occurs.
Case 3: If a 5 0 but b � 0, then multiplication bybi, is a composition, in either order, of a dilationof b and a counterclockwise rotation 90°. Again,this is a special case of a spiral similarity, but alimiting one. (For example, see the result ofmultiplying by i in Exercise 1).
Case 4: If a � 0 and b � 0, the transformation isthe sum of the two images shown in Cases 2 and3. This is a true spiral similarity.
A true spiral similarity occurs when a point ismultiplied by a 1 bi where a � 0 and b � 0.
248
5.
a–b. B 5 2 2 3i, C 5 4, D 5 13c. C 5 (2 1 3i) 1 (2 2 3i) 5 4
D 5
d. Both C and D.C 5
e.
Since a and b are real numbers, 1 2 b2 $ 0 or 1 $ b2.Thus, P 1 Q 5 PQ when (a, b) 5 (1, 1) or (1, 21).
Enrichment Activity 5-6B: Quaternions1. a. 4 1 6i 1 2j 1 3k;
4(1, 0, 0, 0) 1 6(0, 1, 0, 0) 1 2(0, 0, 1, 0) 1 3(0, 0, 0, 1)
b. 27 1 5j 1 8k;27(1, 0, 0, 0) 1 5(0, 0, 1, 0) 1 8(0, 0, 0, 1)
c. 6j 1 9k;6(0, 0, 1, 0) 1 9(0, 0, 0, 1)
d. 23; 23(1, 0, 0, 0)2. a. (8, 0, 2, 3) b. (0, 4, 7, 21)
c. (8, 0, 1, 0) d. (0, 26, 0, 2)3. a. (16, 4, 6, 6) b. (27, 4, 12, 7)
c. (1, 0, 0, 0) d. (8, 0, 7, 9)4. a. 26j b. 210i
c. 24k d. 221e. 21 f. 2kg. k h. j
5. 263 1 37i 1 27j 1 9k6. 2a, a real number8. a. 6 2 7i 2 2j 1 k; 90
b. 3 2 5j 2 2k; 38c. 28i 2 3j; 73d. 25i 1 9j 2 4k; 122
8. 6 solutions; 6i, 6j, 6k
Enrichment Activity 6-2:Arithmetic Sequences1–10. Answers will vary.11. Answers will vary.12. Possible answer: A 5 52, 4, 5, 76 B 5 51, 3, 6, 86
a 5 1 6 "1 2 b2
0 5 a2 2 2a 1 b22a 5 a2 1 b2
P 1 Q 5 PQD 5 (a 1 bi)(a 2 bi) 5 a2 1 b2
(a 1 bi) 1 (a 2 bi) 5 2a
(2 1 3i)(2 2 3i) 5 22 1 32 5 13
yi
xO
A
B
CD
yi
xO
A
BC
D
E
F G
H
I
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Enrichment Activity 7-3:Factoring Expressions with Rational and Negative Exponents
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.13. 14.15. 16.
Enrichment Activity 8-5: Finding e2. 2.7182818233. Yes4. Answers will vary.
Enrichment Activity 8-6:State Population GrowthAnswers will vary by state and with reference used.Sample answers are shown for New York.Population in 1960: 16,782,304Population in 2000: 18,976,457
1. +2,194,1532. About 13.1%3. About 54,8544. y 5 54,854x 1 16,782,304 where x is the number
of years since 19605. a. 19,525,004
b. 20,347,814c. 24,461,864
6. a. 18,976,457 5 16,782,304e40r
b. About 0.31%7. a. 19,595,991
b. 20,528,722c. 25,902,125
8. Possible answer: The exponential model predictslarger populations than the linear model.
9. About 224 years10. About 6.9%
Enrichment Activity 9-7:Reflection and Refraction
1. 35° 2. 17°3. a. 123,917 mi/s
b. 127,572 mi/sc. 139,535 mi/s
(5x16 2 3)(3x
16 2 1)(4b
12 2 1)(2b
12 1 1)
(2y17 1 3)(y
17 2 1)(x
14 1 5)(x
15 2 1)
(x15 1 5)(x
15 2 1)(x
12 1 3)2
5(x2 2 2)x
2(x 2 3)x5
1 2 3c 1 c2
c95
x3 1 1 1 x4
x52
2 1 w5
w121 2 b2
b
x3 2 1x
1 1 xx
35
x13(1 2 x)y
12(y2 1 1)
249
4. 1.654 5. 42° 6. 47° 7. 24°8. 41° 9. 43° 10. 34° 11. 1.179
Enrichment Activity 10-1:Angular Speed and Linear Speed
1. a. (1) 12 hr(2) 60 min or 1 hr
(3) 1 min or hr
b. (1) radians/hr
(2) 2p radians/hr(3) 120p radians/hr
c. (1) p in./hr(2) 16p in./hr(3) 1,200p in./hr
2. a. (1) 2p radians/day(2) radians/hr
b. (1) 12,800p km/day(2) 1,700 km/hr
c. 0 km/any time unit. There is no rotation onthe North Pole because it lies on the axis ofrotation.
3. a. radians/hr
b. 2,400p km/hr (use r 5 6,400 km 1 800 km)4. a. 240p radians/min
b. (1) 3,360p in./min(2) 280p ft/min
c. Yes, 280p ft/min � 879.6 ft/mind. 10 mi/hr
5. a. (1) 480p rad/min(2) 8p rad/sec
b. (1) 1,400p in./min(2) 24p in./sec
6. a. 24p rad/minb. 360p ft/minc. 1,131 ft
Enrichment Activity 10-2:The Angle Between Two Lines
1. a. b. 48° c.
2. a. b. 70° c.
3. a. 1 b. 45° c.
4. a. b. 18° c.
5. a. b. 38° c.
6. a. 82° b. 98°
19p90
79 or 0.7
p10
13 or 0.3
p4
7p8
145 or 2.8
4p15
98 or 1.125
p3
p12
p6
160
14580AKEA.pgs 3/26/09 12:07 PM Page 249
1. See above2.
3. a. b. c. 2p
4. See above5.
6. a. 1.42; b. 21.42; c. 2p
7.
8.
9. a. Max value � 1.3 at 1.047 radiansb. Min value � 21.3 at 5.236 radiansc. 2p
p2 , 3p
2 , 2p
7p4
3p4
21.42; 5p41.42; p4
250
10. a.
b. sin x and sin (x 1 p) have opposite values thatadd to 0 at all values of x.
c. Graph
Enrichment Activity 11-3: Graphing Combined Functions
x 0
sin x 1 cos x 1 1.37 1.42 1.37 1 0.37 0 20.37 21
sin x 2 cos x 21 20.37 0 0.37 1 1.37 1.42 1.37 1
x
sin x 1 cos x 21.37 21.42 21.37 21 20.37 0 0.37 1
sin x 2 cos x 0.37 0 20.37 21 21.37 21.42 21.37 1
2p4p6
7p4
5p3
3p2
4p3
5p4
7p6
p5p6
3p4
2p3
p2
p3
p4
p6
y
1
–1
xp3
2p3
p6
p2
5p6 p 7p
64p3
3p2
5p3
11p6 2p
y
1
–1
xp3
2p3
p6
p2
5p6 p 7p
64p3
3p2
5p3
11p6 2p
14580AKEA.pgs 3/26/09 12:07 PM Page 250
Enrichment Activity 11-4:Polar Coordinates
1. 2. (0, 2)
3. 4.
5. (0, 21) 6. (5, 0)
7. (0, 22) 8.
9. 10.
11. 12.
13. 14.
15. r 5 a 16.
Enrichment Activity 11-8:Graphing Polar EquationsPart I
1.
2. Enlarging a enlarges the size of the petals.If b is odd, the graph has b petals.If b is even, the graph has 2b petals.
3. See the answer to Exercise 2.4. For b odd, graphs involving the sine are
symmetric with respect to the y-axis and graphsinvolving the cosine are symmetric with respectto the x-axis.For b even, both sine and cosine graphs aresymmetric with respect to the x-axis, the y-axis,the origin.
5. See the answer to Exercise 2.
y
x
y
x
y
x
y
x
r 5 2 sin 3u r 5 2 sin 2u
r 5 2 cos 5u r 5 2 cos 4u
r 5 42 cos u 1 3 sin u
A4, 5p6 BA 6, p3 B
(4, p)A5, p2 BA 4!2, 7p
4 BA 3!2, p4 BQ23!3
2 , 232R
Q!32 , 21
2RA22!2, 2!2BQ3
2, 3!32 R
251
Part II6.
7. Each graph is a cardiod (heart) of the same size.Sine graphs are up or down with respect to the y-axis and cosine graphs are right or left withrespect to the y-axis.
8. Sine graphs are symmetric with respect to the y-axis. Cosine graphs are symmetric with respectto the x-axis.
9. Enlarging a enlarges the size of the cardiod.Part III10.
11. When a . 0, the spiral opens right; when a , 0,the spiral opens left.
Enrichment Activity 12-8:Forming Identities
1. f 2. a 3. m 4. e5. b 6. l 7. i 8. n9. k 10. c 11. h 12. o
13. g 14. d 15. j
Bonus:
1. 2. 3. 2 4.
5. 6. 7. 8. !3!34
!33!3
212
!32
2!33
y
x
y
x
r 5 0.2u r 5 20.2u
y
x
y
x
y
x
y
x
r 5 2(1 1 sin u) r 5 2(1 2 sin u)
r 5 2(1 1 cos u) r 5 2(1 2 cos u)
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9. 10. 1 11. 12.
13. 14. 15.
Enrichment Activity 13-4:Solving Trigonometric Inequalities
1.
2.
3.
4. or
5. a.b. Graph y 5 sin x and y 5 cos x and identify
intervals where the graph of sin x is below thegraph of cos x. The solution is 0 # x , or
.
c.
6.
7.
8.
Enrichment Activity 14-4:Heron’s Formula
1.
2.
3.
4.
5 2ab ? c 1 a 2 b
2 ? c 2 a 1 b2
5 fc 1 (a 2 b)g ? fc 2 (a 2 b)g
2ab
5 c2 2 (a 2 b)2
2ab
5 c2 2 (a2 2 2ab 1 b2)
2ab
1 2 cos C 5 2ab2ab 2 a2 1 b2 2 c2
2ab
5 2abs(s 2 c)
1 1 cos C 5 2ab ? a 1 b 1 c
2 ? a 1 b 2 c2
5 a 1 b 2 c2
5 a 1 b 1 c 2 2c2
s 2 c 5 a 1 b 1 c2 2 c
5 2ab ? a 1 b 1 c
2 ? a 1 b 2 c2
5 f (a 1 b) 1 cg ? f (a 1 b) 2 cg
2ab
5 (a 1 b)2 2 c2
2ab
5 a2 1 2ab 1 b2 2 c2
2ab
1 1 cos C 5 2ab2ab 1 a2 1 b2 2 c2
2ab
or 3p2 , x # 7p
4
0 , x # p4 or p2 , x # 3p
4 or p , x # 5p4
p6 , x , 5p
6 or 7p6 , x , 11p
6
p4 # x # 7p
4
p4 # x # 5p
4
5p4 , x , 2p
p4
p4 , 5p
4
7p6 , x , 2pp
6 , x , p
p2 # x # 3p
2
p2 , x , 3p
2
p4 # x , p
2 or 5p4 # x , 3p
2
4!3163
12
12
342
!32
252
5.
6.
7.
8.
(Reject negative root since the sine of any angleof any triangle is always positive.)
9.
10. a. Area 5
b.
11. a. Area 5
b.
12. a. Area 5
b. Area 5
c.
13. a.
b.
Enrichment Activity 14-7:The Law of Tangents
1.tan 12(A 2 B)
tan 12(A 1 B)2 3
10 5
5 60!3 m2
5 !30(18)(10)(2) 5 !10,800
Area 5 !30(30 2 12)(30 2 20)(30 2 28)
5 60!3 m2
Area 5 12(12)(20) sin 1208
/C 5 1208
Area 5 !6(2)(2)(2) 5 !48 5 4!3
12(4)(4) sin 608 5 4!3
12(4)Q4!3
2 R 5 4!3
Area 5 !27(14)(7)(6) 5 !15,876 5 126
12(12)(21) 5 126
5 !15(10)(3)(2) 5 !900 5 30
Area 5 !15(15 2 5)(15 2 12)(15 2 13)
12(5)(12) 5 30
5 !s(s 2 a)(s 2 b)(s 2 c)
5 12ab ? 2ab!s(s 2 a)(s 2 b)(s 2 c)
Area 5 12ab sin C
sin C 5 2ab!s(s 2 a)(s 2 b)(s 2 c)
sin2 C 5 A 2ab B 2
s(s 2 a)(s 2 b)(s 2 c)
1 2 cos2 C 5 A 2ab B 2
s(s 2 a)(s 2 b)(s 2 c)
5 A 2ab B 2
s(s 2 a)(s 2 b)(s 2 c)
(1 1 cos C)(1 2 cos C)
5 2ab(s 2 b)(s 2 a)
1 2 cos C 5 2ab ? c 1 a 2 b
2 ? c 2 a 1 b2
5 c 2 a 1 b2
5 a 2 2a 1 b 1 c2
s 2 a 5 a 1 b 1 c2 2 a
5 c 1 a 2 b2
5 a 1 b 2 2b 1 c2
s 2 b 5 a 1 b 1 c2 2 b
14580AKEA.pgs 3/26/09 12:07 PM Page 252
2. 100°
3.
4.5. A 1 B 5 100, A 2 B 5 39.395
�A 5 69.7°, �B 5 30.3°
6.
7.
8. Find c using the Law of Cosines. Then use theLaw of Sines to find �A or �B.
9. a. 90°b. 1c.
d.
e.
f.
10. �A 5 36.6°�B 5 23.4°
c 5 30.5
Enrichment Activity 15-8:Calculating the Correlation Coefficient
1. 3.5 2. 12.75 3. 2.646 4. 9.6395.
5 22.685 67.48
/B 5 tan21 A 2.56 B/A 5 tan21 A 6
2.5 Btan B 5 2.5
6tan A 5 62.5
!42.25 5 6.562 1 2.52 5 42.25
c 5 6.5
c 5 6sin 67.48
csin 908 5 6
sin 67.48
/B 5 22.68
/A 5 67.48
A 2 B 5 44.8
tan 12(A 2 B) 5 717
/C 5 808c 5 27.3/B 5 30.38b 5 14/A 5 69.78a 5 26
c 5 27.3
c 5 26 sin 808sin 69.78
csin 808 5 26
sin 69.78
A 2 B 5 239.3958
< 20.358
tan 12(A 2 B) 5 2 310 tan 508
253
6. 0.92137. The value from step 6 is very close to the
calculator value of 0.9215.8.
r 5 0.8518
Enrichment Activity 16-3:Chi-Square (x2) Test for Goodness of FitProblem 1
1.
2. 12.863. Yes; 12.86 . 7.81
Problem 21.
sx 5 10.4983, sy 5 15.6365
2x 5 12.25, 2y 5 18.25
xi yi
1 3 20.945 21.011 0.955
2 6 20.567 20.700 0.397
4 20 0.189 0.752 0.142
7 22 1.323 0.960 1.270
Total: 2.764
Qyi 2 2ysy
RQxi 2 2xsx
Ryi 2 2ysy
xi 2 2xsx
xi yi
0 4 21.167 20.911 1.063
2 7 20.976 20.719 0.702
5 16 20.691 20.144 0.099
9 14 20.310 20.272 0.084
12 8 20.024 20.656 0.016
17 25 0.452 0.432 0.195
23 19 1.024 0.048 0.049
30 53 1.690 2.222 3.757
Qyi 2 2ysy
RQxi 2 2xsx
Ryi 2 2ysy
xi 2 2xsx
ExpectedAmount Frequency
$0.50 200 2.88
$1.00 150 0.96
$2.00 100 3.24
$3.00 50 5.78
(Observed 2 Expected)2
Expected
Expected ObservedFace Frequency Frequency
1 100 113 1.69
2 100 88 1.44
3 100 103 0.09
4 100 117 2.89
5 100 95 0.25
6 100 84 2.56
(Observed 2 Expected)2
Expected
14580AKEA.pgs 3/26/09 12:07 PM Page 253
2. 14.923. Yes; 14.92 is greater than the critical value of
11.07. There is sufficient evidence to reject theclaim that the die is fair.
Problem 3Answers will vary.
Enrichment Activity 16-4:Geometric Probability Distribution
1. a. .043b. .037
2. a.b. .076
3. a. .006b. .004
536
254
4. a. (1) Students can assign any four digits tosuccess. Example: let the digits 0, 1, 2, 3represent success.
(2) Students execute the command until a success is found. Theyrecord the number of executionsincluding the success and record theirresults. This is repeated a total of tentimes.
(3) The empirical probability will vary.However, the probability is found bycounting the number of trials wheresuccess occurred on the fourth executionand dividing by the total number of trials(10).
b. Answers will vary. The theoretical probabilityis .0864.
randInt(0, 9)
14580AKEA.pgs 3/26/09 12:07 PM Page 254
Answers for Extended Tasks
255
6. Draw the squares of each walking distance foreach point. Points that are the same walkingdistance from (8, 9) and (6, 5) will be thoselocated on squares of the same size relative toboth points.
7. Draw the line segment connecting (8, 9) and (6, 5). The midpoint of the line segment is (7, 7).The line perpendicular to the segment through(7, 7) is the perpendicular bisector and all pointson the perpendicular bisector of a segment areequidistant from the endpoints of the segment.
8. a. �7 2 2� or �2 2 7� 5 5 unitsb. �5 2 (24)� or �24 2 5� 5 9 unitsc. �x2 2 x1� or �x1 2 x2�
9. a. �20 2 5� or �5 2 20� 5 15 unitsb. �28 2 15� or �15 2 (28)� 5 23 unitsc. �y2 2 y1� or �y1 2 y2�
10. a. �10 2 2� 1 �10 2 4� 5 14 unitsb. �211 2 (25)� 1 �7 2 1� 5 22 unitsc. �x2 2 x1� 1 �y2 2 y1�
Chapter 2Electronic Technician:Applying RationalEquations in the Workplace
a. A series circuit has the resistors positioned toprovide a single path for current flow. A parallelcircuit has the resistors positioned to provide twoor more paths for current flow.
y
xO
5
5
4 3 2 1
4321
(6, 5)
(8, 9)samedistance
samedistance
Chapter 1Going for a Walk
1. 10 units for each route. Routes will vary.Examples:(1, 1) to (7, 1) to (7, 5);(1, 1) to (1, 5) to (7, 5);(1, 1) to (1, 3) to (7, 3) to (7, 5)
2. a. 15 unitsb. 19 unitsc. 21 units
3. a. (0, 6), (1, 5), (2, 4), (3, 3), (4, 2), (5, 1), (6, 0),(7, 1), (8, 2), (9, 3), (10, 4), (11, 5), (12, 6),(11, 7), (10, 8), (9, 9), (8, 10), (7, 11), (6, 12),(5, 11), (4, 10), (3, 9), (2, 8), (1, 7)
b. A squarec. The set of all points outside the squared. The set of all points inside the square
4. a.b. 12 unitsc. The walking distance
5. a. When the points are on the same horizontal orvertical line
b. Never
!72
y
xO
(6, 6)(0, 6)
(6, 0)
(12, 6)
(6, 12)
Answers for Extended Tasks
14580AKET.pgs 3/26/09 12:10 PM Page 255
b.
c.
d. Series Circuit: RT 5 R1 1 R2 1 R3 1 1 RnThe total resistance in a series circuit is the sumof the individual resistances in the circuit.
Parallel Circuit:
The reciprocal of the total resistance in a parallelcircuit is the sum of the reciprocals of theindividual resistances in the circuit.
e. 1. (1) Series circuit(2) R3 5 7,000 ohms
2. (1) Parallel(2) R2 5 30 ohms
3. (1) Parallel circuit(2) R4 5 6,000 ohms
4. (1) Combination series-parallel circuit(2) R6 5 40 ohms
Chapter 3Finding Square Roots Geometrically
Exercise I
1. a. 4 in. b. 3 in.2
2. See construction above.3. See construction above.4. See construction above.5. Yes. Since is the altitude to the hypotenuse of
�ACD, it is the mean proportional between ABDB
A B C
D
M
1RT
5 1R1
1 1R2
1 1R3
1 c 1 1Rn
c
R1 R2 R3
R1
R2
R3
256
and . Since AB 5 1 and BC 5 3, is themean proportional between 1 and 3, or 1 : x 5 x :3. Since the product of the means equals theproduct of the extremes in any proportion, x2 5 3or .
Exercise II1.
2. DB � in. for
DB � in. for
3.
4.
5. � 0.8452
6. � 0.012
Chapter 4The Inverse Variation Hyperbola
Activity 1Students should discover that the product of theforce required to balance the weight and the distancefrom the fulcrum is constant and equal to the weightplaced on the left side peg.
!357
!357
5 67 < 0.8571
5 94 4 21
8
!5 : !7 5 2 416 4 210
16
!721016
!52 416
A B C
D
M1 5
√5
A B C
D
M1 7
√7
x 5 !3
DBBC
14580AKET.pgs 3/26/09 12:10 PM Page 256
257
Activity 2The same relationship should exist.Activity 3Verbal Description: The product of the force, f,exerted and the distance, d, of the spring from thefulcrum is constant and equal to the weight, w, on theleft side of the number balance.Algebraic Description: w 5 fxd or xy 5 c
Activity 4Graphs will differ, but should be a hyperbola (onebranch) in Quadrant I.Activity 5The curves will be the other branch of the hyperboladrawn in Activity 4. This branch will be in QuadrantIII. The curves will have the same equations as statedin Activity 3.Inverse variation is when two quantities change orvary such that their product is a nonzero constant.That is, xy 5 k or y 5 , x not equal to zero.
Chapter 6The Harmonic Series
1. approaches 0.
2.
3. Answers will vary. Some students will think theseries has a sum because the nth termapproaches 0. Others may see that the sums canbe made as large as is required. The series doesnot have a sum.
4. S23
5. S24 5 S16
. 1 1 42
. 1 1 32 1 1
16 1 c 1 116
. 1 1 32 1 1
9 1 c 1 116
Since S8 . 1 1 32 :
5 A 1 1 12 1 c 1 1
8 B 1 A 19 1 c 1 1
16 B
5 1 1 32
. 1 1 22 1 1
8 1 18 1 1
8 1 18
. 1 1 22 1 1
5 1 16 1 1
7 1 18
Since S4 . 1 1 22 :
5 S4 1 15 1 1
6 1 17 1 1
8
5 S8 5 1 1 12 1 1
3 1 14 1 c 1 1
8
S6 5 4920 5 227
60S3 5 116 5 15
6
S5 5 13760 5 217
60S2 5 32
S4 5 2512 5 21
2S1 5 1
1n
kx
6. S2j .The series does not have a limiting sum. Since thepartial sums of the harmonic series have been
shown to be greater than which is unbounded, the partial sums do not approach alimit.
7. 11 8. 31
Chapter 7Holes, Holes, and More Holes:An Exponential InvestigationPart ITask 1
a. Answers will vary, but should be something like:“The total number of holes doubles with eachfold.” or “The number of holes is a power of 2,the power being the number of folds.”
b. 2n
c. h 5 2n
Task 2
a. Answers will vary, but should be something like:“The pattern is similar, but it begins with 21
rather than 20.”b. 2 3 20, , , , ,c.
Task 3
h 5 3(2n)
h 5 2(2n)2 3 252 3 242 3 232 3 222 3 21
1 1j2
. 1 1j2
# of folds 0 1 2 3 4 5
# of holes 1 2 4 8 16 32
# of holes expressedas a power of 2 20 21 22 23 24 25
# of folds 0 1 2 3 4 5
# of holes 2 4 8 16 32 64
# of holes expressedas a power of 2 21 22 23 24 25 26
# of folds 0 1 2 3 4 5
# of holes 3 6 12 24 48 96
# of holes expressed 3320 3321
as a power of 2
3325332433233322
14580AKET.pgs 3/26/09 12:10 PM Page 257
Part II
Answers will vary. Students should observe that the graphs start out very low at about the same point, but riserapidly. As the constant increases for each graph, the graph rises more sharply than the previous one.
h
n0 1
Num
ber
of h
oles
Number of folds
2 3 4 5
12
6
18
24
30
36
42
48
54
60
66
72
78
84
90
96
102
h 5 2n
h 5 2(2n)
h 5 3(2n)
258
Part IIIAnswers will vary. For example, “Yes. It isappropriate because as n, the number of folds,increases, h, the number of punched holes, increasesrapidly.” or “. . . it rises a lot sharper than aquadratic.” or “. . . it increases exponentially.”
Part IVAnswers will vary.
Chapter 8Calculating the Magnitude of an Earthquake:A Mathematical Application
1. 3 2. 2.5 3. 54. 4.1 5. 4
14580AKET.pgs 3/26/09 12:10 PM Page 258
Chapter 9Trigonometry in Aviation
1.
Let the segments of the base of the triangle be represented by m and n.Then: and cot 70° 5 , or
and
Since m 1 n 5 1,000 we will add these 2 equations getting
h 51,000
cot u 1 cot 708
1,000 5 h(cot u 1 cot 708)
m 1 n 5 h cot u 1 h cot 708
n 5 h cot 708m 5 h cot u
nhcot u 5 m
h
1,000 ft
u
Cloudheight
70°
cloud
Observer's eyeParaboliclightsource
Ground
m n
259
2. a. 400 ftb. 730 ftc. 2,300 ft
3. 86° 4. 58°
Chapter 11Temperature
1. 47° 2. 47.5°3–5.
6. Sine curve7. Approximately 365 days8. 23.5
10
10
3
Tem
pera
ture
Month0 2 5 74 6 9 118 10 12
203040506070
9–10. Answers will vary according to data chosenby student.
Chapter 121.
By Formula From Calculator
Angle Sin Cos Sin Cos
1° 0.017460 0.999850 0.0175 0.9998
2° 0.0349 0.9994 0.0349 0.9994
3° 0.0524 0.9986 0.0523 0.9986
4° 0.0698 0.9976 0.0698 0.9976
5° 0.0872 0.9962 0.0872 0.9962
6° 0.1046 0.9945 0.1045 0.9945
7° 0.1219 0.9926 0.1219 0.9925
8° 0.1392 0.9903 0.1392 0.9903
9° 0.1565 0.9877 0.1564 0.9877
10° 0.1737 0.9848 0.1736 0.9848
14580AKET.pgs 3/26/09 12:10 PM Page 259
2. Possible answer: The values were very close.3.
Possible answer: The values are again very close.
Chapter 13Find the Letter: A Trigonometric Puzzle
1. a. sin2 x b. 135°
c. d.
e. f. 210°
g. csc x h.
i. 25° and 225° j.
k.
2–3. I LOVE TRIG. DO YOU?
7p12
6365
!55
p3
cot x1 2 sin xcos x
260
Chapter 14Land for Sale: A Trigonometric Investigation
1. 45,400 ft2
2. 1 acre3. 18 lots4. $5,5005. $99,0006. 52%
Chapter 15Taking a Survey: Designing a Statistical StudyAnswers will vary.
Angle Tan (by formula) Tan (calculator)
1° 0.0175 0.0175
2° 0.0349 0.0349
3° 0.0524 0.0524
4° 0.0700 0.0699
5° 0.0875 0.0875
6° 0.1052 0.1051
7° 0.1228 0.1228
8° 0.1406 0.1405
9° 0.1585 0.1584
10° 0.1764 0.1763
14580AKET.pgs 3/26/09 12:10 PM Page 260
Answers for Suggested Test Items
261
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19. 4 20. 5 21. 322. 8 23. 5524. 25.26. 27.
28. 12 29. 30. 23, 2431. 532. Express: 60 mph, freight: 40 mph33. 4 and 6 34. y . 235. 0 , b , 3 or b . 6 36. x . 5
Bonus: a.
b.
Chapter 31. Rational 2. Rational3. Irrational 4. Rational5. Irrational 6. 25 , x , 57. a # 24 or a $ 10 8. 21 # x # 79. All real numbers 10. 21
11. 4 12. 13. 0.2
14. 15. 16.
17. 18. 19.
20. 21. 22. 129!24!3
11!67!55x6y2#x
y
2a2!77b2!36!5
911
m2 1 m 1 1m3 1 m2 1 2m 1 1
4x2 1 14x(2x2 1 1)
212, 2
x 2 28 (x 2 0, 22)x 1 2
2 (x 2 2, 22)
3ba (a 2 0, b 2 0)x (x 5 0, 1, 21)
5x 2 1x2 2 4x 1 3 ft2 (x 2 1, 3, 23)
9x3x 1 1 Ax 2 0, 13, 21
3 B2y 1 25y2 2 25 (y 2 5, 25)
5x 2 1 (x 2 1, 21)
76 (x 2 21)
43(a 2 3) (a 2 3, 23)
12a (a 2 0)
2(c 1 1)3c (c 2 0, 21, 22)
1x (x 2 0, 1)
23a (a 2 0, a 2 2b)
y 2 6y 2 1 (y 2 1, y 2 4)
2(x 1 7) Ax 223 B
x (y 2 23)
xy2z (x 2 0, y 2 0, z 2 0)Chapter 1
1. 19 2. 5 3. 4 4. 2135. a. All integers
b. All integers n $ 0c. { } or �
6. x 5 7 7. c 5 248. y 5 9 9. x 5 213, 7
10. m 5 24, 1 11. {3, 4, 5, . . . } or k . 212. {2, 3, 4, 5, 6, 7, 8, 9} or 2 # x # 913. n3 2 7n2 1 n 1 2 14. c2d2 1 515. 10x3 2 8x2 1 5x 1 3 16. 22y2 2 2y 1 417. 90 min on math, 62 min on science18. 6 ft and 34 ft19. Kate is 9, her mother is 45.20. {2, 3, 4, 5, 6} or 2 # x #621. {24, 23, 22, 5, 6, 7, . . . } or x , 21 or x . 422. {29, 28, 27, 26, 25, 24, 23, 22, 21, 0, 1, 2, 3, 4,
5, 6} or 29 # x # 623. All integers 24. { } or �25. 22x3 2 12x2 2 18x 26. 24y2 2 4y27. 9x4 2 33x3 1 30x2 28. 22x 1 829. 5x3 2 x2 1 16x 1 1630. a. 2x2 1 3x b. 189 in.2
31. (a 1 b)(5 1 b) 32. (5x 1 6)(2x 2 3)33. (x 2 3y)(x 1 3y) 34. n(4n 2 1)(n 2 1)35. 2x3(2x 2 1)(2x 1 1) 36. (x2 1 4)(x 2 3)37. x 5 26, 5 38. y 5 21, 439. x 5 27, 23 40. x 5 28, 041. 11 and 12 42. 28 in.43. x , 3 or x . 444. {23, 22, 21, 0, 1} or 23 # x # 145. x , 0 or x . 646. {24, 23, 22, 21, 0, 1, 2, 3, 4, 5, 6} or 25 , x , 7
Bonus I:
Bonus II: x , 25 or 23 , x , 2; the product isnegative when an odd number of factors arenegative. If x , 25, all three factors are negative. If xis between 23 and 2, one factor is negative.
Chapter 21. 2. 3. 4. 4
3349
580.416
5 (a 1 b 2 c)(a 2 b 1 c)
5 fa 1 (b 2 c)gfa 2 (b 2 c)g
5 a2 2 (b 2 c)2
a2 2 b2 2 c2 1 2bc 5 a2 2 (b2 2 2bc 1 c2)
Answers for Suggested Test Items
14580AKST.pgs 3/26/09 12:10 PM Page 261
23. 24. 3625. 26. 227. 28. 12
29. 30.
31. 32.
33. a 5 14 34. x 535. b 5 5 36. x 5 1, 2
37.
38. 1.7m, 5.0m, 5.3m 39. Length 5, width 4
Bonus:
Chapter 41. a. {22, 21, 0, 1, 2} b. {1, 2, 3}
c. Yes2. a. {x : 23 # x # 3} b. {y : 23 # y # 3}
c. No; the function is not one-to-one.3. a. {x : x # 9} b. {y : y $ 0}
c. Yes4. a. {x : 0 # x # 3} b. {y : 23 # y # 3}
c. No; the relation is not a function.5. a. {x : 21 # x # 1} b. {y : 21 # y # 1}
c. No; the relation is not a function.6. a. 5 7. a. 21
b. 65 b.8.
9. Domain 5 {all real numbers}Range 5 {26}
10. 1011. a. c 5 3r b. Yes c. $2712. g(x) 5 8(x 2 6) 2 4 5 8x 2 46
214
217
r 5 16 2 8!2
r 58!2 2 16
21
r 58!2
1 1 !2
2r 1 2r!2 5 16!2
r
r
r
r r
r
r√2
#4 1527 . #4 10
27 , so #4 59 .
!4 303
#4 59 5 #4 15
27
!4 303 5
!4 30"4 34 5 #4 30
81 5 #4 1027
!7 2 1
4!x 1 4yx 2 y23 2 2!2
2!5 1 2!23 2 !52
2 1 !59!2 1 627 1 10!2
262
13. p(x) 5 (x 1 6)(x 1 2)(x 2 3) 5 x3 1 5x2 2 12x 2 36
14. a. x2 1 2x 2 8 b. 2x2 1 4 c. 1815. a. 4 b. c. 2116. a.
b.17. a. 3 b. 3
18. 19. y 5 x 2 7
20. y 5 22x 1 1021. a. I b. IV22. a. Yes b. No23. a. (x 1 3)2 1 (y 2 1)2 5 4
b. Center 5 (23,1), radius 5 224. (x 1 4)2 1 (y 2 3)2 5 25 25.26. a. {3, 22} b. 22 , x , 3
c. x , 22 or x . 3
27. a. b. {1, 4}
28. a–b. c. No
29. a. xy 5 6b.
Bonus: x 5 , 2212
y
xO 426 24 222 6
4
2
6
26
24
22
y
xO
a.
b.
y
x21
1
22212223
23456
1 2 3 4 5 6
125 5 2.4
y 5 12x 1 1
2
3x2 2 1(3x 2 1)2 5 9x2 2 6x 1 1
!10
14580AKST.pgs 3/26/09 12:11 PM Page 262
Chapter 51. Since y 5 2(x 2 3)2 2 12, the graph of y 5 x2
must be stretched vertically by a factor of 2 andtranslated 3 units right and 12 units down.
2. 24, 10 3. 26, 3 4. 17i5. 6. 227 7.8. 280 9. 21 1 i 10. 1
11. 2i 12. i 13. 10 2 11i14. 21 2 i 15. 5 2 12i 16. 21 2 2i
17. 5 2 i 18. 7 1 8i 19.
20. 1 6 3i 21. 25 6 2i 22.
23. a. 41 24. a. 9 25. a. 33b. (3) b. (1) b. (3)
26. a. 220 27. a. 0 28. a. 24b. (4) b. (2) b. (4)
29. 1 30. 4 31.
32. x2 1 3x 2 54 5 0 33. x2 2 14x 1 58 5 034. 17, 19 35.
36. , 2 37. 1 6 2i 38.
39. a.
b. (22, 1) (1, 4)40. a.
b. (2, 1)41. a.
b. No real roots(1 1 i, 21 1 i) (1 2 i, 21 2i)
y
xO
y 5 x2 2 x
y 5 x 2 2
y
xO
(2, 1)x2 1 y2 5 5
2x 1 y 5 5
y
xO
(1, 4)
(22, 1)
y 5 x 1 3
y 5 x2 1 2x 1 1
1 6 !312
12 1 12!2 < 28.97 ft
32
12 6 1
2i
16 2 1
6i
40!56i!2
263
42. x , 0 or x . 6 43. 23 # x # 244.
Bonus I: a.b. 4c. Rules for the discriminant apply only
when a, b, c are rational. In this equation,b is irrational.
Bonus II: since c 5 17, 18, 19, or 20 willmake b2 2 4ac negative.
Chapter 61. a. 30, 36, 42 b. an 5 6 1 6(n 2 1) 5 6n2. a. 1, 3, 5 b. an 5 27 1 2(n 2 1) 5 2n 2 93. a. 10, , 13 b.
4. a. b.
5. a. b.
6. a. 21, 1, 21 b. an 5 21(21)n21 5 (21)n
7. 89 8. 270 9. 810. 17 11. 14 12. 44.5113. 80, 105 14. 36, 54, 72, 9015. 6 16. 2117. a. 2 1 5 1 10 1 17 1 26 1 37 b. 9718. a. 12 1 17 1 22 1 27 1 32 1 37 b. 14719. 100 20. 296 21. 2,430
22. 23. 384 24. 1
25. 27,168 26.
27. 18, 36, 72 or 218, 36, 27228. 2105 29. 513 30.
31. 32. 1,310.72 g 33. $3,374.59
34. a. b. 12
35. a. a1 5 1.8, r 5 1.5 b. No sum36. a. b.
Bonus I: 80 ftBonus II: a.
b. No, there is no common ratio.c.
d.
e. 1f.
g. Sn 5 1 2 1n 1 1
1 A 14 2 1
5 B 1 A 15 2 1
6 BS5 5 A1 2 1
2 B 1 A 12 2 1
3 B 1 A 13 2 1
4 B
S10 5 1011, S20 5 20
21
S1 5 12, S2 5 2
3, S3 5 34, S4 5 4
5, S5 5 56
12 1 1
6 1 112 1 1
20 1 130 1 c
257a1 5 5, r 5 22
5
a1 5 6, r 5 12
1278
3,28027
427
22712
an 5 25(2)n2132
5 , 645 , 128
5
an 5 28 A212 B n21
212, 14, 21
8
an 5 4 1 32(n 2 1) 5 3
2n 1 212111
2
P 5 420 5 1
5
x2 2 2!2x 1 1 5 0
212 , x , 1
2
14580AKST.pgs 3/26/09 12:11 PM Page 263
Chapter 71. 3 2. 70 3. 125
4. 4 5. 21 6.
7. 8. 729 9.
10. 11. 12.
13. 14. 15.
16. 17. 18.
19. 20.
21. a–b.
c.
22. 16 23. 24. 1, 3
25. 2 26. 3 27.
28. $3,249.43 29. $8,976.16 30. $12,523.23
Bonus I:Bonus II: They are equal; both equal whenconverted to exponential form.
Chapter 81. y 5 log7 x 2. y 5 2log8 x 3. y 5 6x
4. a. 60 5 x 5. a. 6. a. x4 5 4b. 1 b. 125 b.
7. a. 4x 5 32 8. a. 9. a. x–2 5 0.04
b. b. 23 b. 5
10. 5 11.
12. log 0.01 5 22 13. log x 5 log a 1 2 log b
14.
15.
16.17.
18.
19. 20. N 5!a
(bc)2N 5 ab2
c
log x 5 2 log a 1 log b 1 12 log c
log x 5 23 log a 1 1
3 log blog x 5 2 log a 1 2 log c 2 6 log b
log x 5 log c 1 32 log b 2 5
2 log a
log x 5 12 log a 2 log b 2 3 log c
45 or 0.8
52
3x 5 127
!225
32 5 x
x16
2a
34
12
y 5 A 45 B2x
5 A 54 B x
y
xO
1
21 1
y 5 (54)x y 5 ( 4
5 ) x
5x5!y"4 (3x)3
4x2y73(2x)
121
y665
x7
y43
y2125c3d4
3a2
bx2y24,096
6 164
18 5 0.125
52 5 2.5
264
21. 22.
23. b 2 1 24. 1125.26. a. (1) 5.0969 (2) 2.1761
(3) 5.0969 (4) 2.1761b. 3 log 50 5 log 503;
log (3 3 50) 5 log 3 1 log 5027. 1.53 28. 2.2629. 4.46 30. 20.956631. 2.3036 32. 4.232433. 36 years 34. $6,922.1435. 1.407 36. 2.089
37.
5
38.
39. log 52 5 2 log 5 5 2(0.6990) 5 1.39840. a–b.
c. Reflection in y 5 x
Bonus:
Chapter 91. a. b. c. d.
e. f. g. h.
2. 1,141.7 ft
3. a. b. c.
d. e. f. 3
g. h. i.
j.
4. a. b. c.
d. e. f. 23
!52
3!5 5
3!55
32
2!5 5
2!55
!53
A1, !24 B
!242
2!2321
3
32!2 5
3!242!2
12!2 5
!24
2!23
13
724
725
257
2524
724
247
2425
725
x 5 6x2 5 36
logx (4 ? 9) 5 2logx 4 1 logx9 5 2
y
xO 4
y 5 5x
2 86 1210 1614 2018 22
2468
10121416182022
y 5 log5 x
log 52 5 log 5 2 log 2 5 0.6990 2 0.3010 5 0.398
0.3010 2 0.6990 5 20.398
log 25 5 log 2 2 log 5
57
N 5 a2#3 bc N 5
b!ac
14580AKST.pgs 3/26/09 12:11 PM Page 264
5. IV 6. III 7. II8. 135° 9. 310° 10. 180°
11. 12. 21 13.
14. 15. 21 16. 22
17. 18. 19. 0.7071
20. 20.0523 21. 3.8637 22. 0.373923. 20.6000 24. 1.7013 25. 1.279926. 3.171627. a. 56° b. 55° 499
28. a. 67° b. 66° 569
29. a. 76° b. 76° 089
30. 20.9 31. 225° 32. 2tan 40°33. 2cos 30° 34. cos 20° 35. tan 67.5°
36. 0.18 37. 0.00 38.
39. sin u
Bonus I:Bonus II: �1 5 63.4°, �2 5 26.6°, �3 5 63.4°
Chapter 101. 2. 3.
4. 5. 108° 6. 2540°
7. 315° 8. 80° 9. 2.4 or
10. 4.2 in. 11. 0.4845 12. 1.4506
13. 14. 16 15.
16. 17. 18.
19. 20. 21.
22. 23. 24. 2p3
p32p
6
2p4
p2
p4
cos2 usin2 u
1sin u
1cos2 u
!61 2!22
125
17p6
24p3
p10
5p12
A1, !3B
1sin u
!2!33
2!3
2!32
!22
265
25. 26. 27.
28. 6p 29. 30.
31. 32. 33.
34. 35. 21 36. 35°
37. 30° 38.
Bonus:
Chapter 111. a. 1 b. 2p
c.
2. a. 3 b. p
c.
3. a. 2 b. 4p
c. y
x
2122
12
pp6
3p2
2p3
4p3
7p6
p3
p2
5p6
5p3
11p6 2p
3
23
y
x
2122
12
pp6
3p2
2p3
4p3
7p6
p3
p2
5p6
5p3
11p6 2p
3
23
y
x
20.521
0.51
pp6
3p2
2p3
4p3
7p6
p3
p2
5p6
5p3
11p6 2p
1625
2181
2!32
52
125
2425
2p66p
2
6p22p
6p6
4. a–b.
c. y 5 cos 2x d. y 5 2sin x e. 3
y
x
21
1
pp6
2p3
p3
p2
5p6
5p62
2p32 p
22p32 p
622p
y 5 2sin xy 5 cos 2x
14580AKST.pgs 3/26/09 12:11 PM Page 265
5. a. 1 b. 2p c.
6. a. 2 b. 2p c.
7. a. 1 b. p c.
8. a. b. c.
9. a. or
b.
10. a.
b. or
11.
12. (2`, `)13. The values are increasing to 21.14. 15. 260°16. 60° 17. 230°18. a.
b. 20 sec or min c. 23 ft19. No interval satisfies the condition; since
, if cos x is increasing, sec x must bedecreasing.
Bonus:a.
b. p
c. or for any integer n.
Chapter 121. cot u 2. csc u3. True 4. True5. False 6. True7. a. b. III
8. or 0.75 9. 23!7
834
219
Rx5np2
Tnp, 0
y
x
21
1
pp2
3p2
3p22
p222p
20.5
0.5
2p
sec x 5 1cos x
13
y
x
25
215
51015
210
220
20
!3
Ux : x 2p2 1 npV
y 5 232 cos 32 Ax 2 p
3 By 5 32 cos 32 Ax 1 p
3 By 5 2sin 32x
y 5 2 cos x
y 5 22 sin Ax 2 p2 By 5 2 sin Ax 1 p
2 Bp6
p2
12
2p2
p3
2p4
266
10. 11.
12. 13.
14. 0.96 or 15. 0.8 or
16. 17. 20.28 or
18. 0.936 or 19. 0.75 or
20. 1,210.8 ft
21. a.
✔
b.
22. a.
✔
b.
23. a.
✔
b.
24. a.
✔
b.
Bonus:
Since �x and �y are acute angles, (x 1 y) 5 45°.tan z 5 5 1, so z 5 45°. Finally,x 1 y 1 z 5 45° 1 45° 5 90°.
ss
5 113 1 1
2
1 2 13 ? 1
2tan (x 1 y) 5
tan y 5 s2s 5 1
2
tan y 5 s3s 5 1
3
Let AB 5 BC 5 CD 5 DE 5 s
Ux : x 2np2 V
2 cos xsin2 x
5 2 cos xsin2 x
5? 2 cos xsin2 x
2cos xsin2 xcos2 x
2 sec xtan2 x 5? 2 cos x
sin2 x
2 sec xsec2 x 2 1 5? 2 cos x
sin2 x
sec x 2 1 1 sec x 1 1(sec x 1 1)(sec x 2 1) 5? 2 cos x
sin2 x
1sec x 1 1 1 1
sec x 2 1 5? 2 cos xsin2 x
Ux : x 2 p 1 2pn, p3 1 2pn, 5p3 1 2pnV
sin xcos x 1 1 5 sin x
cos x 1 1
sin x(2 cos x 2 1)(cos x 1 1)(2 cos x 2 1) 5?
sin xcos x 1 1
2 sin x cos x 2 sin x2 cos2 x 1 cos x 2 1 5? sin x
cos x 1 1
sin 2x 2 sin xcos 2x 1 cos x 5? sin x
cos x 1 1
5x : x 2 np6cos2 xsin2 x 5 cos2 x
sin2 x
cos2 x csc2 x 5? cos2 xsin2 x
cos2 x(1 1 cot2 x) 5? cos2 xsin2 x
cos2 x 1 cos2 x ? cot2 x 5? cot2 x
Ux : x 2np2 V
1cos x 5 1
cos x
1sin x ? sin x
cos x 5? 1cos x
csc x ? tan x 5? sec x
34
117125
2725
!210
45
2425
2!6 1 !2
42!32
!6 2 !24
!6 2 !24
14580AKST.pgs 3/26/09 12:11 PM Page 266
Chapter 131. {210°, 330°} 2. {45°, 225°}3. {45°, 135°, 225°, 315°} 4. {30°, 150°, 210°, 330°}5. {120°, 240°} 6. {210°, 270°, 330°}7. {0°, 180°} 8. {30°, 90°, 150°, 270°}9. {0°, 45°, 180°, 225°} 10. {120°, 240°}
11. {90°, 120°, 240°, 270°} 12. {60°, 120°, 240°, 300°}13. 114°, 246° 14. 101°, 259°15. 34°, 82°, 214°, 262° 16. 52°, 128°17. 34°, 180°, 326° 18. 58°, 148°, 238°, 328°19. 80°, 180°, 280° 20. 54°, 147°, 213°, 306°21.
22. 20°, 40°, 70°, 100°, 140°, 160°
Bonus:
Chapter 14
1. 2.
3.4. a. 10 b. 37°5. 30° 6. 167. 126° 8. a 5 12.1, c 5 17.99. 135.9 sq units 10. 661 cm2
11. 162.1 nautical miles 12. 29 ft13. 39.2 in. 14. 47°15. a. 13° b. 480 ft16. a. 2
b. �B 5 53.5°, �C 5 86.5°, c 5 12.4or �B 5 126.5°, �C 5 13.5°, c 5 2.9
Bonus: First find side b using known values:
(We know �C because C 5 180° 2 A 2 B.)Therefore:
5 c2 sin A sin B2 sin (A 1 B)
5 c2 sin B sin A2 sin (180 2 A 2 B)
5 12 A c sin B
sin C B c sin A
Area 5 12bc sin A
bsin B 5 c
sin C S b 5 c sin Bsin C
A0.375, 20.375!3BA26!3, 6BQ3!2
2 , 3!22 R
6 x 5 458, 1358, 2258, 3158
tan x 5 61
�tan x� 5 1
�sin x� 5 �cos x�
0, p6 , 5p6 , p, 7p
6 , 11p6
267
Chapter 151. Census2. a. 21 b. 21.5 c. 22
d. 20 e. 22 f. 6g. 2.83 h. 1.68
3. a. 15 b. 814. a. (1) 2 (2) 2
(3) 2 (4) 1.48b. (1) 2 (2) 1
(3) 0 (4) 2.21c. For Mrs. Alvarez’s data, mean 5 median 5
mode, 70% are within one standard deviationof the mean, 95% within two standarddeviations, and 5% more than two standarddeviations. For Mr. Kazin’s data, the mean,median, and mode are unequal and 85% arewithin one standard deviation of the mean.Mrs. Alvarez’s data more closely resembles anormal distribution.
5. a. 78°F b. 5.546. a. 21.5 b. 2.57. a. The ACT; z-score for SAT � 1.24, z-score for
ACT 5 1.48b. The ACT since her score on the ACT is
farther from the mean than on the SAT.8. a. 34% b. 2.5%9. 76.1% 10. 6,100
11. Close to 21 12. Close to 013. a. y 5 17.676x 1 84.249
sales 5 17.676 (years) 1 84.249b. $261,000c. $703,000; extrapolation
14. a.
b. Exponential; y 5 0.533(1.928)x
c. 196 stores
30252015105
1 2 3 4 5
Year
Stor
es
6 7
50454035
14580AKST.pgs 3/26/09 12:11 PM Page 267
Chapter 161. 720 2. 210 3. 1904. 1 5. 120 6. 1099
7. 27,720 8. 58,464 9. 54
10. 11.
12. 5 35
13. (.3)(.6) 1 (.3)(.4) 1 (.7)(.4) 5 .58 or 58%
14.
15.
16.
17.
18.
19.
20.
< .5370125C0 A 9
10 B 251 25C1 A 1
10 B 1 A 910 B 24
1 25C2 A 110 B 2 A 9
10 B 23
20C13 A 14 B 13 A 3
4 B 7< .0002
10C4 A 23 B 4 A 1
3 B 65
1,12019,683 < .0569
1 2 4C0 A 13 B 0 A 2
3 B 45 1 2 16
81 5 6581
5 C4 ? 15
C1 1 5 C5
20 C55 1
204
Q82RQ
103 RQ5
1R 5 16,800
!34
7!4!3!
5 4 3 312 3 11 5 1
11
10!2!12!4!
12!4! or 19,958,400
268
21. .864322. .556123. x6 1 12x5 1 60x4 1 160x3 1 240x2 1 192x 1 6424. 81x4 1 216x3y 1 216x2y2 1 96xy3 1 16y4
25. 2280a4b3
26.
27. (8n3 2 36n2 1 54n 2 27) in.3
28.
Bonus: 5 people are needed
(greater than 40%)5 1 2 .59049 5 .40951If n 5 5, 1 2 5C0(.1)0(.9)5
(less than 40%)If n 5 4, 1 2 4C0(.1)0(.9)4 5 1 2 .6561 5 .3439Use 1 2 P(exactly 0)
5 220 or 1,048,576
1 c 1 Q2020R10120
(1 1 1)20 5 Q200 R12010 1 Q20
1 R11911 1 Q202 R11812
5t4
27
14580AKST.pgs 3/26/09 12:11 PM Page 268
269
Chapter 61. D 2. C 3. B4. A 5. B 6. C7. D 8. D 9. C
10. B 11. C 12. E13. D 14. A 15. B16. 9 17. 43 18. 6319. 12 20. 70 21. 6422. 150 23. 311 24. 6,860Chapter 7
1. A 2. D 3. D4. E 5. C 6. D7. B 8. E 9. B
10. A 11. B 12. C13. D 14. C 15. D16. 36 17. 1 18. 1219. 13 20. 0 21. 1822. 1 23. 24. 8
Chapter 81. D 2. A 3. D4. B 5. B 6. B7. C 8. A 9. E
10. E 11. C 12. D13. A 14. E 15. D16. 8 17. 1 18. 1819. 0.001 20. or 2.5 21. 1122. 9 23. 4 24. 1Chapter 9
1. B 2. D 3. D4. E 5. E 6. A7. B 8. C 9. D
10. D 11. D 12. A13. A 14. A 15. B16. 100 sq units 17. 6.6 ft 18. 0.81419. 26.4° 20. 67° 21. 1.33322. 60.867 23. 4 24. 0.6Chapter 10
1. E 2. C 3. D4. C 5. A 6. B7. C 8. E 9. A
10. C 11. D 12. C13. C 14. D 15. A16. 9.08 cm 17. 18. 119. 2 20. 30° 21. 10°22. 55° 23. 4 24. 16
9
513
52
73
Chapter 11. C 2. A 3. D4. E 5. D 6. C7. C 8. A 9. B
10. D 11. C 12. E13. B 14. A 15. 1716. 3 17. 11 18. 419. 226 20. 28Chapter 2
1. C 2. C 3. A4. B 5. D 6. A7. D 8. D 9. B
10. C 11. E 12. C13. D 14. E 15. 2816. or 0.2 17. 7 in. 18.
19. 10 mph 20. 21. 19622. 42 boys, 30 girlsChapter 3
1. D 2. A 3. D4. D 5. B 6. B7. C 8. C 9. E
10. D 11. C 12. A13. C 14. C 15. A16. 12 17. 13 18. 519. 4 20. 62 21. 822. 40 23. 0 24. 8Chapter 4
1. C 2. D 3. B4. D 5. A 6. A7. B 8. C 9. B
10. D 11. A 12. B13. A 14. B 15. C16. 16 17. 8 18. 619. 24 20. 6 days 21. 522. 0.011 23. 24. 15
Chapter 51. E 2. A 3. C4. E 5. A 6. B7. E 8. E 9. E
10. C 11. C 12. B13. A 14. C 15. C16. 28 17. 1 18. 62519. 5.5 20. 2.83 21. 822. 25 23. 100 24. 4
23
76
912
15
Answers for SAT Preparation ExercisesAnswers for SAT Preparation Exercises
14580AKSAT.pgs 3/26/09 12:10 PM Page 269
270
Chapter 111. A 2. D 3. D4. E 5. B 6. C7. C 8. D 9. D
10. B 11. C 12. C13. D 14. E 15. C16. 3 17. 30° 18. 219. 0 20. 1.57 21. 1.41
22. 23. 2 24. n
Chapter 121. D 2. C 3. B4. A 5. E 6. A7. D 8. E 9. C
10. E 11. E 12. D13. C 14. E 15. A16. 1 17. 20 18. 6
19. 3 20. 21. 465°
22. 23. 8 24.
Chapter 131. D 2. E 3. B4. C 5. B 6. E7. A 8. B 9. E
10. C 11. D 12. A13. D 14. C 15. B16. 17. 213° 18. 720°19. 45° 20. 45° 21. 45°22. 160.5° 23. 36.4° 24. 3
5p3
964
45
2!10
5
!115
Chapter 141. E 2. D 3. B4. E 5. E 6. C7. C 8. B 9. D
10. D 11. B 12. A13. A 14. D 15. C16. 126.9° 17. 41.4° 18. 4919. 79.8 mi 20. 157 sq units 21. 6.4522. 96° 23. 9.6 24. 25Chapter 15
1. A 2. D 3. D4. E 5. D 6. C7. E 8. C 9. C
10. B 11. A 12. C13. C 14. A 15. C16. 3 17. 1.19 18. 1,90019. 41 20. 164 21. 7122. 500 23. 1,727 24.Chapter 16
1. A 2. C 3. E4. D 5. B 6. B7. B 8. E 9. D
10. E 11. D 12. A13. E 14. C 15. A16. .3087 17. 3,360 18. .327819. .8220 20. 1,260 21. 922. 56 23. 0 # n # 100 24. 7
27 5 0.259
310 5 0.3
14580AKSAT.pgs 3/26/09 12:10 PM Page 270
Answers for Textbook Exercises
271
20. 12n 1 8 5 80, six nights21. 5 1 3h 5 44, 13 hours22. , five plants23. , two hours
1-3 Adding Polynomials (pages 12–13)Writing About Mathematics
1. Yes. If x is negative, will always begreater than x. If x is positive (or zero), 2x 1 1 isalways greater than x.
2. No. Terms in each polynomial may or may nothave like terms in the other polynomial.Furthermore, if like terms have coefficients withequal value but opposite signs, adding them willeliminate a term with that power. Thus, the sumof a trinomial and a binomial may have anywherefrom zero to five terms.
Developing Skills3. 5y 2 13 4.5. 6. 3x7. 8.9. 10.
11. 12.13. 6 14. 415. 25 16. 2317. x , 12, { . . . , 9, 10, 11}18. , {2, 3, 4, . . . }19. , { . . . , 23, 22, 21}20. , {5, 6, 7, . . . }21. 122. , { . . . , 27, 26, 25}Applying Skills23. $2.0024. a. 6x 1 10
b. 4 feet wide and 13 feet long25. 50 cents
1-4 Solving Absolute Value Equations andInequalities (pages 16–17)
Writing About Mathematics1. The absolute value of a number is equal to the
absolute value of its negative.
x # 25
c . 4y # 21y $ 2
2a4 1 a3 2 5a2 1 a 2 14y2 2 3y 2 4x2 1 7x 2 826 2 3b4b2 2 10b2a2b2 1 2
7x2 2 5x 2 45x2 1 x 1 1
�2x 1 1�
5d 1 4 # 1419 1 5d , 49
1-1 Whole Numbers, Integers, and theNumber Line (page 4)
Writing About Mathematics1. Answers will vary. Example: Have Tina count to
three on her fingers, then count to two on herremaining fingers. Show her that if she counts thetotal number of fingers it equals five.
2. Yes. Both sides of the equation refer to the samedistance along the number line.
Developing Skills3. 6 4. 12 5. 56. 5 7. 7 8. 79. 4 10. 0 11. 0
12. 4 13. 25 14. 215. 8 1 (25) 5 316. 7 1 (2(22)) 5 917. 22 1 (25) 5 2718. 28 1 (2(25)) 5 2319. {21, 11}Applying Skills20. 2$2021. a. 2$75
b. 2$2322. 1$100
1-2 Writing and Solving NumberSentences (pages 8–9)
Writing About Mathematics1. Taking an absolute value always yields a positive
number. There is no positive number that can besubtracted from 12 to yield 15.
2. No. Dividing both sides of an inequality by anegative number reverses the direction of theinequality.
Developing Skills3. 7 4. 3 5. 226. 22 7. 4 8. 229. {9, 213} 10. {25, 11}
11. {21, 7} 12. {10, 213}13. , {3, 4, 5, . . . } 14. , {4, 5, 6, . . . }15. , {2} 16. , {4, 5, 6}17. , {21, 0, 1, 2}Applying Skills18. , 49 cents19. 5g 1 3 5 18, three groups
156 2 3g # 9
21 $ b $ 23 , x , 71 , x , 3b $ 4a . 2
Chapter 1.The Integers
Answers for Textbook Exercises
14580AK01.pgs 3/26/09 12:05 PM Page 271
2. Subtract 7 from both sides. The absolute value isthen equal to a negative number, which makesthe solution set empty.
Developing Skills3. {27, 17} 4. {22, 214} 5. {21, 6}6. {23, 7} 7. {1, 7} 8. {3, 24}9. {5, 9} 10. {3, 23} 11. {2, 210}
12. {23, 8} 13. 14. {23, 17}15. x , 29 or x . 9,
{. . . , 212, 211, 210, 10, 11, 12, . . .}16. x , 29 or x . 5, {. . . , 212, 211, 210, 6, 7, 8, . . .}17. , {211, 210, 29, . . . , 23, 22, 21}18. , {0, 1, 2, 3, 4, 5, 6}19. y , 219 or y . 7,
{. . . , 222, 221, 220, 8, 9, 10, . . .}20. or , {. . . , 23, 22, 21, 8, 9, 10, . . .}21. , {22, 21, 0, 1, 2, 3, 4, 5, 6}22. The set of integers23. , {1, 2, 3, 4, 5, 6, 7, 8, 9}24. b , 23 or b . 14, {. . . , 26, 25, 24, 15, 16, 17, . . .}25.26. , {23, 22, 21, . . . , 15, 16, 17}27. {253, 254, 255, 256, 257, 258, 259},28. {150, 151, 152, 153, . . . , 297, 298, 299, 300},
29. , solution 5 ,{172, 173, 174, . . . , 226, 227, 228}
1-5 Multiplying Polynomials (page 21)Writing About Mathematics
1. No. Using FOIL, the answer is.
2. Six. Each term of the trinomial (3) is multipliedby each term of the binomial (2).
Developing Skills
3. 4.5. 6.7. 8.9. 10.
11. 12.13. 14.15. 16.17. 18.19. 20.21. 22. 023. 24. 2525. 3 26. 227. 1 28. 629. 4Applying Skills
30. 2x2 1 4x
z3 2 6z2 1 12z 2 88y2 2 7y 2 10
4b2 1 5b11a 2 122x3 1 5x2 2 7x 2 15y3 2 3y2 1 3y 2 19b2 2 12b 1 4a2 1 6a 1 925b2 2 4a2 2 93x2 2 5x 2 2a2 2 a 2 202x2 1 5x 2 32x2y2 2 4x2y315b2 2 12b29c89c836x2y4212c3d414a8b4
(a 1 3)(a 1 3) 5 a2 1 6a 1 9
172 # c # 228�c 2 200� # 28150 # t # 300
253 # x # 25923 # b # 17�
0 , b , 10
23 , x , 7b $ 8b # 21
21 , y , 7211 # b # 21
�
272
31. a. a 5 c 2 1, b 5 c 2 8b.
c.
1-6 Factoring Polynomials (pages 26–27)Writing About Mathematics
1. Yes. If we multiply these factors back together,we get .
2. No. These factors will yield +4 as the last terminstead of 24.
Developing Skills3. 4.5. 6.7. 8.9. 10.
11. 12.13. 14.15. 16.17. 18.19. 20.21. 22.23. 24.25. 26.27.28.29.30.31.32.33.34.35.36.37.38.39.40.41. 29(x 1 1)(x 1 3)Applying Skills
42. 43.44. 45.
1-7 Quadratic Equations with IntegralRoots (pages 29–30)
Writing About Mathematics1. No. If the product of two expressions is zero, at
least one of the two expressions must be zero.This is not true for other numbers.
(3x 2 1)(x 1 2)(3x 2 1)(3x 2 1)(4x 1 5)(4x 2 5)(4x 1 1)(x 2 2)
3(x 2 3)(x 1 1)y(x 1 4)(x 2 4)(y 1 1)(2y 1 3) 5 21(y 2 3)(y 1 1)(c 1 3)(c 1 1)(z2 2 3)(z 1 3)(z 2 3)2x(2x 2 3)(x 2 1)x(2x 1 3)(x 1 5)(x2 1 4)(x 1 2)(x 2 2)(x2 1 9)(x 1 3)(x 2 3)3(2c 1 1)(2c 2 1)4a(x 1 3)(x 2 2)b(b 1 2)(b 2 2)5(x 2 1)(x 2 2)(a 1 3)(a 1 1)(a 2 1)
(3x 2 2)(3x 2 2)(2y 1 1)(2y 1 1)(6x 2 1)(x 2 2)(5b 1 1)(b 1 1)(2y 2 1)(y 1 3)(3x 1 4)(x 2 3)(x 1 4)(x 1 5)(x 2 3)(x 1 2)(x 1 6)(x 2 1)(x 2 3)(x 2 2)(x 1 3)(x 1 2)(x 1 7)(x 1 1)(y2 2 5)(y 1 1)(x2 2 2)(2x 2 3)(a2 1 3)(a 2 3)(2x 1 3)(y 1 4)(3b 2 4)(b 2 2)(y 2 1)(y 1 1)7(3a2 2 2a 1 1)4a(1 2 3b 1 4a)x2y2(xy 2 2x 1 1)5ab(b 2 3 1 4a)3a2(2a2 2 a 1 3)4x(2x 1 3)
x2 1 (d 1 e)x 1 de
2c2 2 18c 1 655 2c2 2 18c 1 65
a2 1 b2 5 (c 2 1)2 1 (c 2 8)2
14580AK01.pgs 3/26/09 12:05 PM Page 272
2. Yes. If the product of any number of expressionsis zero, at least one of the expressions must bezero.
Developing Skills3. {1, 3} 4. {2, 5} 5. {21, 6}6. {21, 25} 7. {2, 212} 8. {21, 10}9. {21, 4} 10. {3, 210} 11. {22, 3}
12. {23, 4} 13. {1, 7} 14. {3}15. {23, 2} 16. {1} 17. {5}
Applying Skills18. Francis is 11, Brad is 14.19. Length: 30 ft, width: 18 ft20. Width: 8 ft, length: 18 ft21. 9 cm, 12 cm, 15 cm22. 3 seconds
1-8 Quadratic Inequalities (page 35)Writing About Mathematics
1. No. If all three factors are negative, the productwill be negative. Furthermore, if two factors arenegative, the product will be positive.
2. a. Yes. The solution set is 5 , x , 7; thus, anyvalue makes (x 2 7) the negative factor and (x 2 5) the positive factor.
b. No. We can tell for binomial factors of theform (x 1 a)(x 1 b) where a and b are given.However, in other products, such as xy, eitherfactor can be the positive factor.
Developing Skills3. , [
4. x , 26 or x . 1, {. . . , 29, 28, 27, 2, 3, 4, . . .}5. , {1, 2}6. x , 2 or x . 5, {. . . , 21, 0, 1, 6, 7, 8, . . .}7. , {21, 0, 1, 2}8. or ,
{. . . , 24, 23, 22, 10, 11, 12, . . .}9. , {23, 22, 21, 0, 1, 2}
10. x , 1 or x . 5, {. . . , 22, 21, 0, 6, 7, 8, . . .}11. or , {. . . , 22, 21, 0, 2, 3, 4, . . .}12. , {21, 0, 1, 2}13. x , 2 or x . 2, {. . . , 21, 0, 1, 3, 4, 5, . . .}14. The set of integers15. , {21, 0}16. , {23, 22, 21, 0, 1, 2, 3, 4}17. x , 23 or x . 4, {. . . , 26, 25, 24, 5, 6, 7, . . .}
Applying Skills18. {1 ft by 2 ft, 2 ft by 3 ft, 3 ft by 4 ft, 4 ft by 5 ft, 5 ft
by 6 ft, 6 ft by 7 ft}
23 # x # 422 , x , 1
22 , x , 3x $ 2x # 0
24 , x , 3
x $ 10x # 2222 , x , 3
1 # x # 2
23 , x , 22
273
19. {1 ft by 3 ft by 3 ft, 2 ft by 4 ft by 3 ft, 3 ft by 5 ftby 3 ft, 4 ft by 6 ft by 3 ft}
Review Exercises (pages 37–38)1. 22x 2. 2a 1 123. 23d 1 7 4.5. 6.7. 8.9. 4 10. 0
11. 12.13.14. or 15.16.17.18.19.20.21.22.23.24.25. 2926. 1327. x . 4, {5, 6, 7, . . . }28. , {21, 0, 1, 2, 3}29. {2, 27}30. {6, 28}31. y , 21 or y . 2, {. . . , 24, 23, 22, 3, 4, 5, . . .}32. x , 25 or x . 21, {. . . , 28, 27, 26, 0, 1, 2, . . .}33. {4, 5}34. {5, 7}35. 26 , x , 21, {25, 24, 23, 22}36. x , 25 or x . 7, {. . . , 28, 27, 26, 8, 9, 10, . . .}37. , {0, 1, 2, 3, 4, 5}38. x , 23 or x . 0, {. . . , 26, 25, 24, 1, 2, 3, . . .}39. , {1, 2, 3}40. x # 22 or x $ 1, {. . . , 24, 23, 22, 1, 2, 3, . . .}41. An absolute value cannot be equal to a negative
number.42. Width: 12 cm, length: 32 cm43. Width: 8 ft, length: 30 ft44. 10 in., 24 in., 26 in.45. a. 96 feet
b. 1 second and 4 seconds
Exploration (page 38)1. 6; 28; 496; 8,1282. All Euclidean perfect numbers have as a
factor. Since k is always a positive integer greaterthan 1, Euclidean perfect numbers will bemultiples of 2.
2k21
1 # x # 3
0 # x # 5
21 # x , 4
(5x 2 3)(x 1 5)5(a2 1 b2)(a 1 b)(a 2 b)x(x 2 2)(x 2 1)2(x 2 3)(x 2 6)(x 1 1)(x 2 1)(x 1 1)(x 2 1)(x 2 1)(x 1 1)(x 1 5)3(y2 1 2)(y 2 4)(c2 1 4)(c 1 2)(c 2 2)10a(b 1 2)(b 2 2)5x(x 1 1)(x 2 4)
3(a 2 5)23(a 2 5)(a 2 5)2(x 1 1)(x 1 3)
y2 2 4y2x2
x2 2 x 2 114d2 1 19cd 2 3c222a2 2 2ax2 1 7x 2 203b2 2 25b 1 45
14580AK01.pgs 3/26/09 12:05 PM Page 273
3. The possible units digits of any power of 2 are {2, 4, 6, 8}.Given the units digit of any 2k21, the units digit of(2k 2 1) 5 2(2k21) 2 1.If 2k21 5 2, then the units digit of (2k 2 1) 5 3.If 2k21 5 4, then the units digit of (2k 2 1) 5 7.If 2k21 5 6, then the units digit of (2k 2 1) 5 1.
274
If 2k21 5 8, then the units digit of (2k 2 1) 5 5.However, integers with units digit 5 (other than 5itself) are not prime, so (2k 2 1) � 5 and 2k21 � 8.The product of 2 3 3 and of 6 3 1 is 6. Theproduct of 4 3 7 ends in 8. Therefore, aEuclidean perfect number N 5 2k21(2k 2 1) musthave a units digit of 6 or 8.
Chapter 2.The Rational Numbers
2-1 Rational Numbers (page 43)Writing About Mathematics
1. a. The coin is called a quarter because it is 25 outof 100 cents, one-fourth the value of a dollar.
b. A quarter of something is equivalent to one-fourth of its total value. Since the totalnumber of minutes in an hour and cents in adollar differ, one-fourth of those values willalso differ.
2. The additive inverse makes the sum of the twonumbers equal zero. The multiplicative inversemakes the product of the two numbers equal toone.
Developing Skills
3. 4. 5.6. 7. 18. 0.166 . . . 5 09. 0.222 . . . 5 0
10. 0.7142857142 . . . 5 011. 0.133 . . . 5 012.13. 14. 15.
16. 17. 18.
19. 20. 21.
22.
2-2 Simplifying Rational Expressions (pages 47–48)
Writing About Mathematics1. Abby is wrong. 3x is not a common factor of the
numerator and denominator, and cannot becanceled out.
2. No. It is true for all values except where thedenominator is zero .
Developing Skills3. a 5 0 4. c 5 05. a 5 0, b 5 0 6. x 5 25
Aa 5 32 B
744
322
2645
56
47300
437
411
29
23
18
0.8750.13
.714285.2.16
18
272
127
83
7. a 5 8. b 5 2, 239. c 5 0, 1 10. x 5 21, 0, 6
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21.
22. 23.
24. 25.
26. 27.
28. 29.
30.
2-3 Multiplying and Dividing RationalExpressions (pages 52–53)
Writing About Mathematics1. No. Joshua needed to write the reciprocal of the
second fraction before attempting to cancel outany common factors.
2. Yes. He divided the terms separately, which isacceptable based upon the commutativeproperty of multiplication.
Developing Skills
3. 4.
5. 6.
7. 8.
9.
10.
11. 12.
13. 14.
15. 16. 16 (a 2 0)4 (b 2 0, c 2 0)
8 (a 2 0)53
252x (x 2 23, 0, 3)1 (a 2 0, 22)
2(a 2 1)3(a 1 4) (a 2 24, 22, 4)
3y2 2 3y (y 2 23, 0, 3)
a2 1 10a6 (a 2 0, 10)3
5 (b 2 21)
23 (a 2 0)1
10 (x 2 0, y 2 0)
328 (a 2 0)1
2
2 5b 1 4 (b 2 24, 4)
22x 2 3 (x 2 3)1
b 1 2 (b 2 2, 22)
a 1 1 (a 2 1)13a 1 3 (a 2 21, 1)
5(y 2 2)y 1 2 (y 2 22)x 2 4
x 1 5 (x 2 3, 25)
a 2 2 (a 2 22)23 (a 2 25)
13 1 2xy2 Ax 2 0, y 2 0, xy2
2 232 B
cc 1 2 (c 2 0, 22)2
3 2 4d Ad 2 0, 34 B4a 2 2b
3a (a 2 0, b 2 0)3y 1 1
2y (y 2 0)
2a 1 43a (a 2 0)3
4c3 (c 2 0, d 2 0)
2b3 (b 2 0)
4yx (x 2 0, y 2 0)
ab2 (a 2 0)3
5
72
14580AK01.pgs 3/26/09 12:05 PM Page 274
17. 18.
19. 20.
21. 22.
23.
24. 25.
26. 27.
28.
29. 30. 3x (x � 0, 1, 2)
2-4 Adding and Subtracting RationalExpressions (pages 56–57)
Writing About Mathematics1. No. It is also undefined when a 5 1.2. Yes. He formed a correct LCD and added.
Developing Skills
3. x 4.
5. 6. or
7. 8.
9. 10.
11. 12.
13. 14.
15.
16.
17.
18.
19.
20.Applying Skills
21. a. 22. a.b. 2 b. x 1 1
23. a. 24. a.
b. b.
2-5 Ratio and Proportion (pages 60–61)Writing About Mathematics
1. Yes. Interchanging the means or extremes of aproportion maintains the equality of theproportion.
2. Yes. One is added to each side of the equation,which maintains the equality.
x2
(x 1 1)(x 1 1)3x
(x 2 1)(x 2 1)
4x2 1 6x(x 1 1)(x 1 2)
2x 1 6x 2 1
18x 1 203
4x2 1 2x
2x (x 2 0, 2)
1a(a 2 2) (a 2 22, 0, 2)
1(2a 2 1)(a 1 2) Aa 2 2 2, 2
12, 23 B
1x 2 2 (x 2 2)
2b 1 12(b 2 1) (b 2 1)
x2 1 3x 2 4x(x 1 2) (x 2 0, 2 2)
1 1 xx (x 2 0)2a2 2 3
2a (a 2 0)
10y 2 12y (y 2 0)3x 1 2
x (x 2 0)
11a 1 26a (a 2 0)x 1 12
12x (x 2 0)
3a 1 8040a (a 2 0)
7y6
2a 1 920
2a 2 920
10x21
25x2 1 25x (x 2 0)
6(b 1 2)2 (b 2 0)
(x 2 1)(x 1 4)x (x 2 0, 1)
12 (a 2 22, 0, 2)9 (x 2 21, 0, 1)
14
a2 2 a2 (a 2 21)
(2x 1 7)2(x 2 1) Ax 2 1, 272 B
a 1 54a (a 2 0, 23)4
b (b 2 0, 23)
1w 1 1 (w 2 21, 0, 1)1 (c 2 3)
22y 1 1 Ay 2 21
2, 0, 12 B34x (x 2 0, 2)
275
Developing Skills3. 3 : 2 4. 3 : 2 5. 1 : 66. 1 : 5 7. 2 : 3 8. 2 : 39. 1 : 3 10. 2 : 7 11. 2
12. 7 13. 7 14.15. 7 16. {23, 2} 17. {0, 5}18. {21, 4} 19. {210, 2}Applying Skills20. 16 inches wide by 20 inches long21. 28 inches long by 12 inches wide22. 15 games23. 33 members24. $75 and $5025.26. 4 cups of solution and 28 cups of water
2-6 Complex Rational Expressions (pages 63–64)
Writing About Mathematics
1. {21, 0, 1}. An expression of the formis undefined when either x, y, or z
is zero.2. No. When , the denominator will equal
zero, which would make the fraction undefined.Developing Skills
3. 4 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18. (y � 28, 23, 0)
19. 20.
21. 22.
23. 24.
2-7 Solving Rational Equations (pages 69–70)
Writing About Mathematics1. Yes. Samantha multiplied both sides by the LCD,
which is a valid way to solve this equation.2. Brianna is correct. A rational equation has a
variable in one or more denominators.
21 (b 2 22, 0, 2)3a 1 24a Aa 2 0, 53 B
174a (a 2 0)1
2x (x 2 0, 21)
25(a 1 3)
a (a 2 0, 3)21 (a 2 0, 1)
y 2 2y 1 8
b 1 2b 2 1 (b 2 21, 0, 1)
3x 2 5 (x 2 0, 3, 5)
(a 1 7)(a 2 2) (a 2 0, 2, 7)
5x6 (x 2 0)
2(2y 1 1)y Ay 2 0, 12 B
12 Ay 2 21
2 B3b (b 2 0, 1)
2(b 1 1) (b 2 0, 1)22 (d 2 0, 1)
1a (a 2 21, 0)1
2
4 (x 2 0)12
110
d2
2 5 2
1x 4
yz 5 1
x ? zy
214 cups
13
14580AK01.pgs 3/26/09 12:05 PM Page 275
Developing Skills3. 32 4. 8 5. 86. 70 7. 12 8. 209. 80 10. 20 11. 8
12. 10 13. 12 14.15. 16. 17. 318. {5, 27} 19. 4 20.Applying Skills21. Week 1: Joseph worked 8 hours and Nicole
worked 12.Week 2: Joseph worked 12 hours and Nicoleworked 24.
22. 5 mph23. 40 mph then 50 mph24. Price: $1.25, 6.6 lb the 1st week, 7.6 lb the
2nd week
2-8 Solving Rational Inequalities (pages 73–74)
Writing About Mathematics1. The number line must be separated by the
solutions to the equation as well as the values atwhich any of the rational expressions areundefined.
2. . Since the numerator will be positivefor any nonzero rational number and undefinedat 0, the expression is negative for all x , 0.
Developing Skills3. a , 224 4. y , 85. 6. d , 27. 8. 0 , x , 19. 0 , y , 4 10.
11. 12.
13. 14.
Review Exercises (pages 75–76)1. 2. {21, 0, 1}
3. 0 4.
5.
6.
7.
8. 9.
10. 11.
12. 13.
14. 15. 2x 1 1x (x 2 0, 1)x 2 1
x 2 2 (x 2 22, 2 1, 2)
a 1 4 (a 2 4)2a 1 2a2 1 2a (a 2 22, 2 1, 0)
12 (b 2 0)d 2 3
d (d 2 26, 0)
y 1 6y 1 3 (y 2 23, 3)2a 1 1
a22 1
(a 2 21, 1)
a2
5a 1 25 (a 2 25, 2 2, 0)
32x(x 2 4) (x 2 24, 0, 4)
1320a (a 2 0)
2a3b (a 2 0, b 2 0).416
27
25 , a , 2127 , x , 25
x , 0 or x . 12
53 , x , 4
a , 22 or a . 21a . 153
5
b . 25
5x:x , 06
73
U22, 32V12
285
276
16.
17.
18.
19.
20. 21.
22. 6 23. {0, 8}
24. 2 25.
26. 27.
28. 27 boys, 30 girls29. Week 1: 15 cans, week 2: 12 cans, $0.70 per can30. 60 mph then 45 mph31. 16 ft by 13 ft and 14 ft by 13 ft
Exploration (page 76)1.
2.
3. a. b.
c. d.
e.
Cumulative Review (pages 77–78)Part I
1. 4 2. 1 3. 3 4. 2 5. 1 6. 1 7. 2 8. 2 9. 3
10. 1Part II11. Answer: 2 # x # 5
7 2 2x # 3 and 7 2 2x $ 23x $ 2 x # 5
�7 2 2x� # 3
12 1 1
9
12 1 1
3 1 18
12 1 1
12
12 1 1
513 1 1
15
or
34 5 1
2 1 14
23 5 1
2 1 16
112
23 5 1
3 1 14 1
5 1n
5 n 1 1n(n 1 1)
1n 1 1 1 1
n(n 1 1)
x , 212 or x . 2 9
20x , 0 or x . 2
U22, 32V
203
111
a(a 2 5)a 2 4 (a 2 24, 0, 4)
1x 1 6 (x 2 26, 6)
ab (a 2 0, b 2 0, a 2 2b)
23
14580AK01.pgs 3/26/09 12:05 PM Page 276
12.
Part III
13.
14. Width: 6 m, length: 15 m
w 5 230, w 5 6. Reject negative value.(w 1 30)(w 2 6) 5 0w2 1 24w 2 180 5 0
w2
2 1 12w 2 90 5 0
w Aw2 1 12 B 5 90
l 5 w2 1 12
5 x3 (x 2 21, 1)
5 5(x 1 1)
(x 2 1)(x 1 1) ? x(x 2 1)
15
5x 1 5x2 2 1 ? x
2 2 x15
5 26(a 1 5)(a 1 3) (a 2 25, 2 3, 3)
5 3(a 1 3) 2 3(a 1 5)
(a 1 5)(a 1 3)
5 3a 1 5 2 3
a 1 3
5 3a 1 5 2 a 2 3
5 ? 15(a 2 3)(a 1 3)
3a 1 5 2 a 2 3
5 4 a2 2 915
277
Part IV
15. Answer: x , 21 or
The solutions for the corresponding equationsare and 21.When x , 21, the inequality is true.When , the inequality is false.
When , the inequality is true.16. Diego traveled at 60 mph and then at 20 mph.
x 5 60
60x 5 1
30x 1 30
x 5 1
5 110x3
30x 1
x . 72
21 , x , 72
72
(2x 2 7)(x 1 1) . 02x2 2 5x 2 7 . 0
2x2 2 5x . 7
x . 72
Chapter 3. Real Numbers and Radicals
3-1 The Real Numbers and Absolute Value(page 83)
Writing About Mathematics
1. No. The expression can be written as the ratioof two integers, so it is rational.
2. No. Maria’s inequality is a false statement. If sheapplied the rule “�x� . k, then x . k or x , 2k,”she would get 22x 1 5 . 3 or 2x 2 5 , 23.
Developing Skills3. rational 4. irrational5. irrational 6. irrational7. rational 8. irrational9. irrational 10. rational
11. rational 12. irrational13. rational 14. irrational
In 15–26, answers will be graphs of number lines.15. 27 , x , 7 16. a $ 8 or a # 217. y . 2 or y , 27 18. 21 # b # 219. a , 29 or a . 21 20. 24 , x , 221. x . or x , 22. { } or �23. all real numbers 24. x 5
25. all real numbers 26. all real numbers
Applying Skills27. 70° # t # 220°28. 282 ft # h # 20,320 ft
245
215
25
A 154 B
3-2 Roots and Radicals (pages 87–88)Writing About Mathematics
1. a. Yes. Since the product of an even number ofeither positive or negative factors is positive,the radical will have both a positive and anegative root.
b. Yes. The product of an odd number of positivefactors is positive, and the product of an oddnumber of negative factors is negative. Thus aradical with a real, nonzero radicand and anodd index will have only one real root.
2. a. Yes. This is true only when a $ 0. The otherequal factor is .
b. The statement is true for a $ 0. When a , 0, ahas no square roots in the set of real numbers.
Developing Skills3. rational 4. irrational5. neither 6. rational7. rational 8. rational9. rational 10. rational
11. 4 12. 64 13. 2414. 25 15. 13 16. 20.217. 60.8 18. 1.2 19. 320. 2 21. 25 22. 2523. 5 24. 21 25.26. 27. 28.29. 0.1 30. 0.4 31. x3
212
2327
6
25
2 !a
14580AK01.pgs 3/26/09 12:05 PM Page 277
32. 10c2 33. 0.5x 34.35. 36. 20.1y 37. 138. x2 39. x $ 2 40. x # 341. x $ 23 42. x $ 25 43. x 5 6944. a 5 614 45. b 5 610 46. y 5 613Applying Skills
47. cm � 3.74 cm48. ft � 23.32 ft49. 15 in.50. 6 in.51. 13 ft
3-3 Simplifying Radicals (pages 93–94)Writing About Mathematics
1. is the negative of the square root of 36,which is a real number, simplifying to 26.is the square root of a negative number and is notreal.
2. Negative. If a is negative, 28a3 will be positiveand its cube root will be also positive. Thenegative sign in front makes the wholeexpression negative.
Developing Skills
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26.
27. 28.
29. 30.
31. 32.
33. 34.
35. 36.
37. 38. 2xy!4 2xa2 "4
2b2cbc
a3!3 910!3c
8x!2x100!10x
11b2!2ab10
!22
!25
25y2!10xy
310xy4!2xy2a
9b2!3ab
!30b4b2
!10a6
!15b5b3
!3xy2y
!6xy6xy
a5b!5ab
a2
2 !2a"69y3
b7!3b2a2
5
2a2 "4 3ab35xy2
"3 3x2
2a!3 5a2!3 3
2!3 23b2!2ab
44x2y3!3xy50y!2x
6y2!5y7c2!2
2b!2b4!2
5!22!3
!2362!36
!544!14
2b2
6
10a
278
Applying Skills39. cm 40. in.41. m 42. ft43. ft 44. m45. 6y2 units46. The longest diagonal of the trunk is or
approximately 40.05 inches. Thus, everything butthe walking stick will fit.
3-4 Adding and Subtracting Radicals (pages 97–98)
Writing about Mathematics
1. Yes, for x . 0. (3x)2 5 9x2, and 5 3x. Her
substitution is correct.2. No. We do not add radicands.
, which is not equal to.
Developing Skills
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26.
27. 28.
29. 30.31. 32.
33. 34.
35.36. 37.
38. 39.
40. 41.
42.
Applying Skills43. in.14!3
!53
!6!22
!33x!x
15x!2xb!a3a!7 2 3a!5 or 3aA!7 2 !5B
23b!2!2a
5!y8!x!4 37!3 2
3!3 212!6
3!5 2!10109!3 2 2!6
6a2b!2b 1 26!3y 1 y2
22x!65a!5 2 5!2a
!2xx4!7
3!66!55
22x3!511b2!6b
12x!2x4c2!2c
5a3!2a5y!6x
5b2!116a!10
!5y6!2
4!79!3
2!56!2
!64 5 8!16 1 !48 5 4 1 4!3
"9x2
!1,604
xy2!55!65!1312!36!24!13
14580AK01.pgs 3/26/09 12:05 PM Page 278
44. ft
45. a. cm
b. cm
46. a. 14 in.b. 1 14 in. or in.
47. a. mb. m
3-5 Multiplying Radicals (pages 100–101)Writing About Mathematics
1. Yes. is a positive real number.2. Yes. We can simplify by dividing the exponent of
the radicand by the index.Developing Skills
3. 4 4. 15
5. 9 6.
7. 8.
9. 10.
11. 12. 12
13. 27 14. 20
15. 2x2 16.
17. 18.
19. 20.
21. 22. 2
23. 24. 3
25. 26.
27. 28.
29. 30.
31. 32.
33. 49 2 5b 34.
35. 36. 6 2 36c2
37. a2 2 b 38.
39. 40.
41.
Applying Skills
42. 4,608 m2 43. 120 ft2
44. in.3
45. a. ft b. ft c. 3 ft2
46. m2p(4 1 xy5 1 4y2!xy)
6 1 2!62!6!5
22 2 !5
26 2 6!79 1 6b!5ab 1 5ab3
4 2 2!3
236 2 !6
2x2 2 3x!4 3y 1 !3y
21 2 4!5y 2 5y9 1 10!2b 1 2b
22 1 2!56xy2 1 6y!3xy
5a 2 3!5a12!2 1 4
!5 2 5!22!2 1 2
3a2!3 5
a6!3a
x2!x35a
3
x3y2!32y2!5
4ab!b
4!10
2!72!2
6!526!5
4!6
!2
13!1034!10
14A1 1 !2B14!2
12!2
5!2
25!3
279
3-6 Dividing Radicals (pages 103–104)Writing About Mathematics
1. No. Jonathan’s error was treating the
denominator of as . does not simply
further.
2. Answers may vary. Example: ,
which is rational. , which is irrational.
Developing Skills
3. 2 4. 5
5. 3 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 1 22. 5
23. 24.
25. 26.
27. 28.
29.
Applying Skills
30. ft31. a. cm
b. 6 cm
3-7 Rationalizing a Denominator (pages 107–108)
Writing About Mathematics
1. a. If Juan writes 7 as , the fraction becomes
. This is equivalent to , which
simplifies to .b. No. Juan’s procedure cannot be applied to
because 5 is not a factor of 49.2. Brittany took the fraction at face value and
multiplied by the conjugate of the denominator.Justin saw that the denominator factored to
. 2 is a factor of the numerator, so thefraction is equivalent to .2
1 1 !2
2A1 1 !2B
72!5
12!7
12#49
7!492!7
!49
2!6
52!7
8!x 1 !4 40
2!3 ww!c
1a"4 a3 or "
4a3
a!3 9 1 "3 12x2
!5 1 61 1 12!6
2 1 !3!2 2 3!10
2 1 4!6c!2
!3a!7y
y
!3xx
3!2b2b
2!6xy3y
a!142
2!33!3
5!22!2xx
a!10
!6!3 5 !2
!27!3 5 !9 5 3
!102!2!10
2
14580AK01.pgs 3/26/09 12:05 PM Page 279
Developing Skills
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14. 1
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26.
27. 28.
29. 30.
31. 32.
33. 34.
35.
36.
37.
38.
39. a.
b. 1.632993162c. 1.632993162
40. a.
b. 2.154700538c. 2.154700538
41. a.b. 5.464101615c. 5.464101615
42. a.b. 15.93725393c. 15.93725393
43. 44.
45. 46. !8 2 !5!3 2 1
!3!2
8 1 3!7
2!3 1 2
2!3 1 33 or 23!3 1 1
23!6
3x2 2 3x!2 1 (5x2 2 10)!xx(x2 2 2)
2a!ab 1 2b!abab or 2!ab
ab (a 1 b)
3!x 1 (36 2 x)!6 2 183x 2 108
2y!x 1 2x!yxy
a 1 2!aa 2 4
4!z 2 32z 2 64
2x 1 5y 2 7!xyx 2 y
(a 1 2) Ab 1 !2Bb2
2 2
27 1 7!522
11 2 2!109
17 1 2!79
8 1 5!27
10y!y 2 2!5y5y2 2 1
5x 1 2!5x5x 2 4
3 2 !3!2 1 1
3!7 2 63!5 1 6
16 2 4!79
3!5 2 34
6 1 2!33
!3 2 12
5 1 !223
3 2 !54
16!10
16!62
3!6
!343!3
!612!3
23!32!2
12!10!3
3
280
Applying Skills
47. ( ) in. 48. ft, ft, ft
3-8 Solving Radical Equations (pages 112–113)
Writing About Mathematics1. There is no real number for which the square
root is negative. If x $ 23 the radicand will notbe negative, so there will be a solution in the setof real numbers.
2. No. Once we square the equation it has two realroots. One is the root of the given equation andthe other is the root of .
Developing Skills3. 25 4. 49 5. 96. 36 7. 16 8. 329. 4 10. 8 11. 4
12. 44 13. 4 14. 215. 2 16. 25 17. 418. 4 19. 5 20. 221. 21 22. 5 23. 5, 424. { } (no solution)25. 5 26. 3 27. 1
28. 3 29. 30. 8
31. 210 32. 22 33. 23434. 18 35. 4 36. {0, 8}37. 5 38. 5Applying Skills39. 8 units each40. Width 5 2, length 5 141. a.
b.
Review Exercises (pages 114–116)
In 1–8, answers will be graphs of number lines.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16. 12
17. 75 18. 4 1 3!2
5!6 1 2!5
4!221!3
!515!3
!328!2
0 , x , 221 # x # 215
x . 1 or x , 213x # 235 or x $ 21
5
x , 22 or x . 4.5210.5 # x # 11.5
26.5 # x # 2.5214 , x , 3
4
5!10AB 5 BC 5 2!10, AC 5 !10
U1, 23V
!15 2 7x 5 1 2 x
2!232!23
2!26!5 2 6
14580AK01.pgs 3/26/09 12:05 PM Page 280
19. 20.
21. 22. 22
23. 24.
25. 26.
27. 28.
29. 30.
31. 32.
33. 34.
35. 36.
37. 38.
39. 40. a
41. 42. 13
43. 5 44. 18
45. 14 46. {3, 4}
47. 5 48. 25
49. {22, 3} 50. ft
51. a. 4 mb. m
52. Show that these values satisfy the PythagoreanTheorem:
True for all values of because of theradical.
53. a. ftb. ft
Exploration (page 116)Answers will vary.
Cumulative Review (pages 116–118)Part I
1. 4 2. 3 3. 44. 3 5. 1 6. 37. 4 8. 4 9. 4
10. 1Part II
11.
5 2a 1 3
5(a 1 4)(a 2 4)(a 1 3)(a 2 4) ? 2a
a(a 1 4)
5 a2 2 16a2 2 a 2 12 ? 2a
a2 1 4a
a2 2 16a2 2 a 2 12 4 a2 1 4a
2a
52 2 22!324 2 12!3
x . 0
x2 1 2x 1 1 5 x2 1 2x 1 1
x2 1 A!2x 1 1B2 5 (x 1 1)2
4 1 4!2
12 1 7!3
x2
16 1 a 2 8!a16 2 a
3 1 !x9 2 xb2!3 9
4a23a3!3 2
x2y2!y4x 2 x!4 36
12x2!x!abb
x!4 12x7b!2b
8!a4!3 1 313
4!2 2 22!5 1 1
!25 1 3!3
21
6!2 1 6020 2 20!2
281
12.
✔
Part III13.
14. No. correct 5 2(no. incorrect) + 4Let no. incorrect 5 xNo. correct 5 2x + 4
Plugging back in, Tyler answered 16 questionscorrectly.
Part IV15. Use D 5 RT, where D is distance, R is speed, and
T is time. Then .Let x 5 Rachel’s speed on local streets, and 2x beher highway speed, both in mph.
Rachel travels 30 mph on local streets and 60mph on the highway.
16. let w 5 the width in yards. The length is thereforel 5 4w 1 2.Use A 5 lw.
w(4w 1 2) 5 30
4w2 1 2w 2 30 5 0
2w2 1 w 2 15 5 0
(2w 2 5)(w 1 3) 5 0
w w 5 23Reject negative value.
The garden is yards wide and 12 yards long.212
5 212
x 5 3016x 5 480
8x 5 16
60
2 1 6x 5 16
60
2x 1 12
2x 5 1660
T 5 DR
x 5 62x 5 128x 5 6x 1 12
2x 1 4x 5 8
3
2413 , b , 1
3
2133 , b , 1
3
213 , 3b , 127 , 3b 1 6 , 7
�3b 1 6� , 7
5 5 5
2 A 14 B 1 9
2 5? 5
2 A212 B 2
2 9 A212 B 5? 5
Check: x 5 212
x 5 212
(2x 1 1)(x 2 5) 5 0
2x2 2 9x 2 5 5 0
2x2 2 9x 5 5
x 5 5
✔5 5 52(25) 2 45 5? 5
2(5)2 2 9(5) 5? 5Check: x 5 5
14580AK01.pgs 3/26/09 12:05 PM Page 281
4-1 Relations and Functions (pages 126–127)
Writing About Mathematics
1. {(x, y) : x 5 y2} is not a function because for allvalues of x . 0 there are two distinct y-values,whereas is a function because forevery value of x $ 0 there is exactly one realnumber that is the square root. No two pairs havethe same first element.
2. No, because not every positive integer has anintegral square root. The range contains non-integer values.
Developing Skills3. a. function; no two pairs have the same first
elementb. {1, 2, 3, 4}c. {1, 4, 9, 16}
4. a. not a function; points (1, 21) and (1, 1) havethe same first element
b. {0, 1}c. {21, 0, 1}
5. a. function; no two pairs have the same firstelement
b. {22, 21, 0, 1, 2}c. {5}
6. a. functionb. {all real numbers}c. {y : y $ 21}
7. a. not a functionb. {all real numbers}c. {y : y # 21 or y $ 1}
8. a. not a functionb. {x : x $ 22}c. {all real numbers}
9. a. functionb. {all real numbers}c. range: y 5 2
10. a. functionb. {x : 23 # x # 3}c. {y : 0 # y # 4}
11. a. functionb. {x : 1 # x # 6}c. {y : 0 # y # 2.5}
12. a. {all real numbers}b. The function is not onto since the range is
{2183}.13. a. {all real numbers}
b. The function is onto since the range is equalto the domain.
5(x, y:!x 5 y6
282
14. a. {all real numbers}b. The function is not onto since the range is
{y : y $ 0}.15. a. {all real numbers}
b. The function is not onto since the range is.
16. a. {x : x $ 0}b. The function is onto since the range is equal
to the domain.17. a. {all real numbers}
b. The function is not onto since the range is {y : y $ 0}.
18. a. {x : x � 0}b. The function is onto since the range is equal
to the domain.19. a. {x : x # 3}
b. The function is onto since the range is equalto the domain.
20. a. {x : x . 21}b. The function is not onto since the range is
{y : y . 0}.21. a. {all real numbers}
b. The function is not onto since the range is {y : 0 , y # 1}.
22. a. {x : x � 1}b. The function is onto since the range is equal
to the domain.23. a. {x : x � 3}
b. The function is not onto since the range is {y : y , 1}.
Applying Skills24. a. {(x, y) : y 5 x(6 2 x)}
b.
c. {x : 0 , x , 6}25. a. {(x, y) : y 5 10x}
b. (0, 0), (1, 10), (2, 20), (3, 30), (4, 40), (5, 50),(6, 60), (7, 70), (8, 80)
c. {0, 1, 2, 3, 4, 5, 6, 7, 8}d. {0, 10, 20, 30, 40, 50, 60, 70, 80}
x 0 1 2 3 4 5 6 7 8 9 10
y 0 5 8 9 8 5 0 27 216 227 240
yx
25210215220225230235240
5
1 2 3 4 5 6 7 8 9 10O
Uy : y # 14V
Chapter 4. Relations and Functions
14580AK02.pgs 3/26/09 12:05 PM Page 282
4-2 Function Notation (pages 128–129)Writing About Mathematics
1. f and g are the same function, since evaluating gat x yields the same value as f.
2. f and g are not the same function. For example,note that f(3) 5 9, while g(3) 5 g(1 1 2) 5 3.
Developing Skills3. a. f(x) 5 x 2 2 4. a. f(x) 5 x2
b. 3 b. 255. a. f(x) 5 �3x 2 7� 6. a. f(x) 5 5x
b. 8 b. 257. a. 8. a.
b. 2 b.
9. 10 10. 10 11. 1212. 4 13. 2 14. 115. a. 2 b. 23 c. 22 d. 2Applying Skills16. a. t(a) 5 0.08a
b. {a : a $ 0}c. $0.40d. $1.32
17. a.
b. $300c. 3,000 muffins
4-3 Linear Functions and Direct Variation(pages 133–135)
Writing About Mathematics
1. Yes. , when a . 1, is equivalent tof(x) 5 ag(x), and ag(x) is the graph of g(x)stretched vertically by a factor of a.
2. Yes. Directly proportional means the ratio of r : sis constant. Every direct variation of twovariables is a linear function that is one-to-one.
g(x) 5 1af(x)
y
xO
2900
1,000
28002700260025002400230022002100
100200300400500600700800900
2,000 3,000
25
f(x) 5 2xf(x) 5 !x 2 1
283
Developing Skills3. a. {1, 2, 3, 4} 4. a. {0, 2, 4, 6}
b. {4, 7, 10, 13} b. {8, 6, 4, 2}c. yes c. yes
5. a. {2, 3, 4, 5, 6} 6. a. {0, 21, 22, 23, 24}b. {7} b. {3, 5, 7, 9, 11}c. no c. yes
7. no 8. no 9. no10. yes 11. no 12. no13. a. graph 14. a. graph
b. yes b. noc. yes c. yes
15. a. graph 16. a. graphb. yes b. noc. yes c. yes
17. a. graph 18. a. graphb. yes b. noc. yes c. yes
19. a.
b. yesc. yes
20. a.
b. yesc. yes
21. no22. yesApplying Skills23. 24. 25.
26. 27.
28. a. g(t) 5 80 1 25tb. {t : 0 # t # 420}c. {g(t) : 80 # g(t) # 10,580}d. yese. No. The ratio g(t) : t is not constant.
gm 5 25
gk 5 1,000
if 5 12d
t 5 35cn 5 6
y
x1
O1
y
x1
O1
14580AK02.pgs 3/26/09 12:05 PM Page 283
Hands-On Activity 11–2.
3. The graph of g(x) is the graph of f(x) reflected inthe y-axis.
4. The graph of h(x) is the graph of f(x) reflected inthe x-axis.
5.
6. The graph of p(x) is the graph of f(x) stretchedvertically by a factor of 3.
7. The graph of 2p(x) is the graph of f(x) reflectedin the x-axis and stretched vertically by a factorof 3.
8. f(x) 5 x 1 1(1) Graph of g(x) 5 f(2x) 5 2x 1 1(2) Graph of h(x) 5 2f(x) 5 2x 2 1(3) The graph of g(x) is the graph of f(x)
reflected in the y-axis.(4) The graph of h(x) is the graph of f(x)
reflected in the x-axis.(5) Graph of 3f(x) 5 3x 1 3(6) The graph of p(x) is the graph of f(x)
stretched vertically by a factor of 3.(7) The graph of 2p(x) is the graph of f(x)
reflected in the x-axis and stretched verticallyby a factor of 3.
y
xO1
1
y
x1O21
h(x)
g(x)
284
Hands-On Activity 21–2.
3. The graph of p(x) is the graph of f(x) shifted aunits to the left.
4. The graph of p(x) is the graph of f(x) shifted aunits to the right.
5–6.
7. The graph of f(x) 1 a is the graph of f(x) shifteda units up.
8. The graph of f(x) 2 a is the graph of f(x) shifteda units down.
9.
10. The graph of af(x) is the graph of f(x) stretchedvertically by a factor of a.
11. y
xO21 1
y
xO
1
1
y
xO
21 1
f(x) 1 2
f(x) 2 4
y
xO1
1
g(x)
h(x)
14580AK02.pgs 3/26/09 12:05 PM Page 284
12. The graph of 2f(x) is the graph of f(x) reflectedin the x-axis or y-axis.
13.
14. The graph of g(x) is the graph of f(x) reflected inthe x-axis or y-axis.
4-4 Absolute Value Functions (pages 138–139)
Writing About Mathematics1. Yes. Each y then corresponds to exactly one x.2. Yes. By definition, f(x) 5 2 2 x when x # 2 and
f(x) 5 x 2 2 when x . 2.Developing Skills
3. (0, 0) 4. (24, 0)5. (14, 0) 6. (5, 0)
In 7–10, the answer to part a is a graph.
7. b. {y : y $ 0} 8. b. {y : y $ 0}9. b. {y : y $ 1} 10. b. {y : y $ 23}
Applying Skills11. h(x) 5 �2 2 x�12. a. m(x) 5 �x 2 150�
b.13. a–c. Graphs
d. The graph of y 5 �x� 1 a is the graph of y 5 �x�shifted vertically by the amount �a�. When a . 0, the shift is upward. When a , 0, theshift is downward.
14. a–c. Graphsd. The graph of y 5 �x 1 a� is the graph of y 5 �x�
shifted �a� horizontally. When a . 0, the shift isto the left. When a , 0, the shift is to the right.
15. a–b. Graphsc. The graph of y 5 2�x� is the graph of y 5 �x�
reflected in the x-axis.16. a–c. Graphs
d. The graph of y 5 a�x� is the graph of y 5 �x�stretched or compressed vertically. When a . 0, the graph is stretched vertically. When a , 0, the graph is compressed vertically.
h(x) 5�x 2 150�
65
y
xO21 1
285
17. a.
b. {28, 2}c. x , 28 or x . 2
18. a.
d. 28 , x , 2b. {2, 6}c. 2 , x , 6d. x , 2 or x . 6
4-5 Polynomial Functions (pages 147–149)Writing About Mathematics
1. Yes. Tiffany is correct. The graph never crosses ortouches the x-axis.
2. This function has two roots; x 5 2 is a doubleroot.
Developing Skills3. a. no real roots 4. a. 21
b. {3} b. {all real numbers}c. no c. yesd. no d. yes
5. a. {0, 2} 6. a. {21, 3}b. {y : y $ 21} b. {y : y # 4}c. no c. nod. no d. no
7. a. 1 8. a. 0b. {y : y # 0} b. {all real numbers}c. no c. yesd. no d. yes
9. a. {22, 0, 2} 10. a. {23, 21, 1, 3}b. {all real numbers} b. {y : y $ 21}c. no c. nod. no d. no
11. a. {22, 0, 3}b. {all real numbers}c. yesd. no
12. 23 , x , 1 13. 23 # x # 2114. x , 21 or x . 2 15. { } or �16. x , 1 or x . 5 17. 24 # x # 1
y
xO
121
y
xO
1
21
14580AK02.pgs 3/26/09 12:05 PM Page 285
Applying Skills18. a. 12 2 x
b. y 5 x(12 2 x) or y 5 12x 2 x2
c.
d. length 5 width 5 619. a. 20 2 x
b. y 5 or y 5
c.
d. Each leg measures 10 feet.20. p(x) 5 (x 1 4)(x 1 2)(x 2 3) or
p(x) 5 x3 1 3x2 2 10x 2 2421. a–c. Graphs
d. y 5 x2 1 a is the graph of y 5 x2 shiftedvertically by the amount �a�. When a . 0, thegraph of y 5 x2 is shifted upward. When a , 0,the graph of y 5 x2 is shifted downward.
e. T0,a22. a–c. Graph
d. y 5 (x 1 a)2 is the graph of y 5 x2 shiftedhorizontally by the amount �a�. When a . 0,the shift is to the left. When a , 0, the shift isto the right.
e. Ta,023. a–b. Graphs
c. The graph of y 5 2x2 is the graph of y 5 x2
reflected in the x-axis.d. Ry=0
24. a–c. Graphsd. y 5 ax2 is the graph of y 5 x2 stretched
vertically.e. y 5 ax2 is the graph of y 5 x2 compressed
vertically.
y
xO
555045403530252015105
25 5 10 1520 25
10x 2 x2
212x(20 2 x)
y
xO
363330272421181512
963
2 4 6 8 10
286
25.
4-6 The Algebra of Functions (pages 153–155)
Writing About Mathematics1. No. �2 2 x� and �x 2 2� are both always $ 0 so
their sum is always $ 0. Their sum is equal to 0only when x 5 2.
2. If g(x) 5 x 1 1, 2g(x) 5 2x 1 2 � g(2x) 5 2x 1 1.If f(x) 5 x, 2 f(x) 5 2x 5 f(2x) 5 2x.
Developing Skills3. a. {0, 1, 2, 3, 4, 5}
b. {1, 2, 3, 4, 5, 6}c. {1, 2, 3, 4, 5}d. {(1, 23), (2, 1), (3, 7), (4, 15), (5, 25)}e. {1, 2, 3, 4}f.
4. a. f(0) 5 1b. x 5 1c–e.
5. a. f(0) 5 22b. x 5 62c–e.
6. a. f(0) 5 1b. { } or �c–e. y
x1O
2f(x)
f(x) 1 2
2f(x)
21
y
x21 O
2f(x)
f(x) 1 2
2f(x)
21
y
x1O
1
2f(x)
f(x) 1 2
2f(x)
U A1, 14 B , A2, 43 B , A3, 92 B , (4, 16)V
A2 b2a, 4ac 2 b2
4a B
14580AK02.pgs 3/26/09 12:05 PM Page 286
7. a. f(0) 5 21b. x 5 61c–e.
8. a. x2 2 2x 1 4 9. a.
b. {all real numbers} b. {x : x � 0}10. a. x 11. a.
b. {x : x � 0} b. {x : x � 2}12. a. x2 2 6x 1 12 13. a. 2x3 2 4x2
b. {all real numbers} b. {all real numbers}Applying Skills14. a. c(x) 5 10x 1 2
b. t(x) 5 0.15(10x) 5 1.5xc. e(x) 5 10x 1 2 1 0.15(10x) 5 11.5x 1 2d. e(3) 5 $36.50
15. a. c(x) 5 8.50xb. s(x) 5 0.50x 1 2c. t(x) 5 8.50x 1 0.50x 1 2 5 9x 1 2d. t(5) 5 $47.00
16. Answers will vary. Example:Let f(x) 5 x 1 1; then show f(x) 1 f(x) 5 2f(x).
f(x) 1 f(x) 5 x 1 1 1 x 1 1 5 2x 1 22f(x) 5 2(x 1 1) 5 2x 1 2
This result is true in general since doubling afunction is the same as adding it to itself.
4-7 Composition of Functions (pages 159–160)
Writing About Mathematics1. Yes. f(x) 5 x2 evaluates to (a 1 1)2 at x 5 a 1 1
by definition.2. fg(x) is the product of the functions f(x) and g(x).
f(g(x)) is function composition.Developing Skills
3. 6 4. 10 5. 212 6. 287. 45 8. 1 9. 24 10. 0
In 11–18, the answer to part d is a graph.
11. a. h(x) 5 8x 1 4b. {all real numbers}c. {all real numbers}
x2
4 2 2x
x2 2 1x
y
xO
2f(x)
f(x) 1 22f(x)
287
12. a. h(x) 5 3x 2 1b. {all real numbers}c. {all real numbers}
13. a. h(x) 5 4 1 x2
b. {all real numbers}c. {y : y $ 4}
14. a. h(x) 5 x2 1 8x 1 16b. {all real numbers}c. {y : y $ 0}
15. a. h(x) 5 xb. {x : x $ 0}c. {y : y $ 0}
16. a. h(x) 5 2�2 1 x�b. {all real numbers}c. {y : y # 0}
17. a. h(x) 5 �5 2 x�b. {all real numbers}c. {y : y $ 0}
18. a. h(x) 5 xb. {all real numbers}c. {all real numbers}
19. f(g(x)) 5 �x 1 3�g(f(x)) 5 �x� 1 3
20. f(g(x)) 5 �2x�g(f(x)) 5 2�x�
21. f(g(x)) 5 �2x 1 3�g(f(x)) 5 2�x� 1 3
22. f(g(x)) 5 �5 2 x�g(f(x)) 5 5 2 �x�
23. Exercise 20: g(x) 5 2x24. p(q(5)) 5 2
q(p(5)) 5 425. Answers will vary. For example, f(x) 5 2x and
g(x) 5 x 1 1.Applying Skills26. a. c(x) 5 1.08x
b. d(x) 5 x 2 10c. 5 1.08x 2 10.80
5 1.08x 2 10No. applies sales tax to the discountedprice, while discounts the price aftersales tax has been applied. It makes sense to use
on in-store discounts. It makes sense touse on discounts applied after purchase,for example, a $10 mail-in rebate.d. The function to use depends on whether the
tax is applied to the full price or to thediscounted price.
27. a. 5 4(0.55x2 1 1.66 x 1 50) 2 160 5 2.2x2 1 6.64x 1 40
b. 128.2 chirps per minute
n + f
d + c(x)c + d(x)
d + c(x)c + d(x)
d + c(x)c + d(x)
14580AK02.pgs 3/26/09 12:05 PM Page 287
4-8 Inverse Functions (pages 166–167)Writing About Mathematics
1. Yes. The graph of the inverse is the reflection ofthe graph of f over the line y 5 x.
2. No. When the domain of an absolute valuefunction is the set of real numbers, the function isnot one-to-one and has no inverse function.
Developing Skills3. 3 4. 5 5. 22 6. 87. 12 8. 26 9. 26 10.
11. Yes; f21 5 {(8, 0), (7, 1), (6, 2), (5, 3), (4, 4)}12. Yes; f21 5 {(4, 1), (7, 2), (10, 1), (13, 4)}13. Yes; f21 5 {(8, 0), (6, 2), (4, 4), (2, 6)}14. No. The function is not one-to-one, so it has no
inverse function.15. No. This relation is not a function.16. Yes; f21 5 {(x, y): for 2 # x # 27}17. a.
b. domain of f 5 domain of f21
5 {all real numbers}range of f 5 range of f21 5 {all real numbers}
18. a. g21(x) 5 x 1 5b. domain of g 5 domain of g21
5 {all real numbers}range of g 5 range of g21 5 {all real numbers}
19. a. f21(x) 5 3x 2 5b. domain of f 5 domain of f21
5 {all real numbers}range of f 5 range of f21 5 {all real numbers}
20. a. f21 5 x2
b. domain of f 5 domain of f21 5 {x : x $ 0}range of f 5 range of f21 5 {y : y $ 0}
21.
22. g21(x) 5 7 2 x. Yes, a function can be its owninverse.
23. No. y 5 x2 is not one-to-one if the domain is theset of real numbers.
24. y
xO
211
f21 5 U(x, y) : y 5 x5V
f21(x) 5 x 1 34
y 5 !x 2 2
!2
288
25.
26.
Applying Skills
27.
Since f(f21(x)) 5 f21(f(x)) 5 x, the functions areinverses.
28. a. Graphb. domain 5 {x : x $ 24}
range 5 {y : y $ 2}c. domain 5 {x : x $ 2}
range 5 {y : y $ 24}d. The domain of the function is the range of the
inverse and the range of the function is thedomain of the inverse.
4-9 Circles (pages 172–173)Writing About Mathematics
1. No. A circle does not pass the vertical line test.2. In center-radius form, the constant term is the
square of the radius, and this cannot be negative.Developing Skills
3. a. x2 1 y2 5 4b. x2 1 y2 2 4 5 0
4. a. x2 1 y2 5 9b. x2 1 y2 2 9 5 0
5. a. x2 1 y2 5 16b. x2 1 y2 2 16 5 0
6. a. (x 2 4)2 1 (y 2 2)2 5 1b. x2 1 y2 2 8x 2 4y 1 19 5 0
7. a. (x 1 1)2 1 (y 2 1)2 5 16b. x2 1 y2 1 2x 2 2y 2 14 5 0
f21(f(x)) 5 0.2532x0.2532 5 x
f(f21(x)) 5 0.2532 A x0.2532 B 5 x
f21 5 x0.2532
y
x
21O
1
y
x1 O1
14580AK02.pgs 3/26/09 12:05 PM Page 288
8. a. (x 2 6)2 1 (y 2 5)2 5 100b. x2 1 y2 2 12x 2 10y 2 39 5 0
9. a. (x 2 6)2 1 (y 2 13)2 5 169b. x2 1 y2 2 12x 2 26y 1 36 5 0
10. a. x2 1 (y 2 1)2 5 17b. x2 1 y2 2 2y 2 16 5 0
11. x2 1 y2 5 1612. (x 2 2)2 1 (y 2 3)2 5 113. (x 2 1)2 1 (y 1 1)2 5 914. (x 1 2)2 1 (y 2 3)2 5 415. (x 2 1)2 1 (y 1 1)2 5 2516. x2 1 (y 1 1)2 5 417. (x 1 1)2 1 (y 2 3)2 5 918. (x 2 1)2 1 (y 2 1)2 5 1319. (x 1 1)2 1 (y 1 1)2 5 1320. a. x2 1 y2 5 25
b. (0, 0)c. 5
21. a. (x 2 1)2 1 (y 2 1)2 5 9b. (1, 1)c. 3
22. a. (x 1 1)2 1 (y 2 2)2 5 4b. (21, 2)c. 2
23. a. (x 2 3)2 1 (y 1 1)2 5 16b. (3, 21)c. 4
24. a. (x 1 3)2 1 (y 2 3)2 5 12b. (23, 3)c.
25. a. x2 1 (y 2 4)2 5 16b. (0, 4)c. 4
26. a. (x 1 5)2 1 (y 2 2.5)2 5 63.25b. (25, 2.5)
c. 5
27. a.
b.
c.Applying Skills28. a. Graph
b. Yes. The cube can easily pass through the archbecause its sides are shorter than the radius ofthe arch.
c. Yes. If the prism is placed in the center of thearch so that its base is 8 feet, it will have justunder 7 feet of clearance to pass under the arch.
29.30. Width 5 , length 5 8!54!5
10!2
32!2
A212, 32 B
Ax 1 12 B 2
1 Ay 2 32 B 2
5 184
!2532!63.25
2!3
289
31. Answers will vary: any three equations of theform (x 2 2)2 1 (y 2 2)2 5 r2 with three differentpositive values for r.
Hands-On Activity1. (1, 3)2. Slope of ; the slope of the line
perpendicular to is undefined.3. x 5 14. (1) (0, 0)
(2) Slope of ; slope of the lineperpendicular to
(3) y 5 x5. C(1, 1)6. CP 5 CQ 5 CR 5
7. Equation: (x 2 1)2 1 (y 2 1)2 5 20P is on the circle: (5 2 1)2 1 (3 2 1)2 20
16 1 4 5 20 ✔Q is on the circle: (23 2 1)2 1 (3 2 1)2 20
16 1 4 5 20 ✔R is on the circle: (3 2 1)2 1 (23 2 1)2 20
4 1 16 5 20 ✔
4-10 Inverse Variation (pages 177–178)Writing About Mathematics
1. No. The function f cannot be represented in theform f(x) 5 y 5 .
2. In direct variation, both quantities increase ordecrease by the same factor. In inverse variation,as one quantity increases by a factor a, the otherquantity decreases by the factor .
Developing Skills
3. xy 5 2 or
4. xy 5 6 or
5. xy 5 28 or
6. inversely 7. directly8. directly 9. directly
10. inversely11. neither directly nor indirectly12. directlyApplying Skills13. The width of rectangle ABCD is equal to half the
width of rectangle EFGH.14. He can ride to school in one-third the time it
takes him to walk.15. a. Yes. D 5 RT. As rate increases, time traveled
decreases, when distance is constant.b. 45 mph
y 5 28x
y 5 6x
y 5 2x
1a
anxn 1 an21xn21 1 c 1 a0
5?
5?
5?
2!5
QR 5 1QR 5 21
PQPQ 5 0
14580AK02.pgs 3/26/09 12:05 PM Page 289
16. Initial trip: 55 mph for 3 hrsReturn trip: 33 mph for 5 hrs
17. 1st time: 16 cans at $1.50 per can2nd time: 15 cans at $1.60 per can
Review Exercises (pages 180–183)1. Not a function since (1, 1) and (1, 21) have the
same first element.2. Function. Each x of the domain has only one
y-value.3. Not a function since for most values of x in the
domain there are two distinct y-values in therange.
4. Function. Each x value of the domain has onlyone y-value in the range.
5. Function. Each x value of the domain has onlyone y-value in the range.
6. a. yes 7. a. nob. yes b. no
c. yes; c. no
8. a. no 9. a. yesb. no b. yesc. no c. yes;
10. a. no 11. a. yesb. no b. yesc. no c. yes; y 5 x2 1 4, x $ 0
12. 1 13. {21, 3}14. {22, 4} 15. {22, 21, 1}16. 1 17. {23, 5}18. x 5 27 and x 5 119. {(0, 7), (1, 8), (2, 9), (3, 10), (4, 11)}20. {(0, 5), (1, 2), (2, 21), (3, 24), (4, 27)}21. {(0, 6), (1, 15), (2, 20), (3, 21), (4, 18)}
22.
23. f(g(x)) 5 2x 1 6 24. f(g(x)) 5 �4x 2 1�g(f(x)) 5 2x 1 3 g(f(x)) 5 4�x� 2 1
25. f(g(x)) 5 (x 1 2)2 26. f(g(x)) 5g(f(x)) 5 x2 1 2 g(f(x)) 5
27. f(g(x)) 5 28. f(g(x)) 5
g(f(x)) 5 g(f(x)) 5
29. f(g(x)) 5 x 30. f(g(x)) 5 xg(f(x)) 5 x g(f(x)) 5 x
31. x2 1 y2 5 932. (x 2 3)2 1 (y 2 3)2 5 9
50x2 1 3
2x
12x
2 1 3x2
5!x 2 3!5x 2 3
U(0, 6), A1, 53 B , A 2, 45 B , A3, 37 B , (4, 29 B V
y 5 !3 x
y 5 x 2 34
290
33. (x 2 3)2 1 y2 5 2534. x2 1 (y 1 1)2 5 1635. (x 1 1)2 1 (y 1 1)2 5 536. (x 1 2)2 1 (y 2 2)2 5 837. (21, 22) and (3, 2)
Check (21, 22):
✔
✔
Check (3, 2):
✔
✔
38. y 5 (x 2 4)2 2 2 39. y 5 3�x�40. y 5 22x 2 3 41. y 5 2�x 1 1� 1 342. a. f21(x) 5 (x 2 8)
b. Yes, since f is one-to-one.43. a.
b. Yes, since f is one-to-one.44. a.
b. No, f is not one-to-one.c. {x : x $ 0}
45. a. f21(x) 5 x2
b. Yes, since f is one-to-one.
Exploration (page 183)1. The base is a circle. The two cut edges are circles.2. The cut surfaces are ellipses.3. The curved portion of the edges is a parabola.4. The shape is a hyperbola.5. The shape is a pair of intersecting lines.
Cumulative Review (pages 184–185)Part I
1. 3 2. 2 3. 24. 2 5. 3 6. 47. 4 8. 2 9. 4
10. 4Part II11. 2(x 1 1)2 (x 2 1)
12. 3 1 !53 2 !5 ? 3 1 !5
3 1 !5 5A3 1 !5B2
9 2 5 57 1 3!5
2
f21(x) 5 !x
f21(x) 5 3x
12
2 5 22 5? 3 2 1y 5 x 2 1
10 5 10(1)2 1 (3)2 5? 10
(x 2 2)2 1 (y 1 1)2 5 10
22 5 2222 5? 21 2 1
y 5 x 2 110 5 10
(23)2 1 (21)2 5? 10(x 2 2)2 1 (y 1 1)2 5 10
14580AK02.pgs 3/26/09 12:05 PM Page 290
Part III13. Answer: 23 , x , 5
Let (x 1 3) , 0:x 1 3 , 0 x 2 5 . 0
x , 23 x . 5Solution: { }Let (x 1 3) . 0:
x 1 3 . 0 x 2 5 , 0x . 23 x , 5
Solution: 23 , x , 5Combine the two solutions.
14. D 5 22h 5 24, E 5 22k 5 6F 5 h2 1 k2 2 r2 5 4 1 9 2 16 5 23x2 1 y2 1 Dx 1 Ey 1 F 5 0
x2 1 y2 2 4x 1 6y 2 3 5 0
(x 1 3)(x 2 5) , 0x2 2 2x 2 15 , 0
291
Part IV15. a.
b. x 5 1c. (1, 29)d. x 5 20.5 and x 5 2.5
16. ;
undefined for a 5 0, a 5 �1.
2aa 2 1? a
2
a2 5 2a2 1 2aa2 2 1 5
2a(a 1 1)(a 1 1)(a 2 1) 5
2 1 2a
1 2 1a2
yx1O
1
Chapter 5. Quadratic Functions and Complex Numbers
5-1 Real Roots of a Quadratic Equation(pages 192–193)
Writing About Mathematics
1.
2. Yes. The resulting equation is equivalent to theoriginal. The new equation can be solved bycompleting the square.
Developing Skills3. 19 5 (x 1 3)2 4. 116 5 (x 2 4)2
5. 11 5 (x 2 1)2 6. 136 5 (x 2 6)2
7. 12 5 2(x 2 1)2 8.
In 9–14, part b, answers will vary.
9. a. y
xO21
1
194 5 Ax 2 3
2 B 2
x 5 1, 212
0 5 8(x 2 1)(2x 1 1)0 5 16x2 2 8x 2 80 5 2x2 2 x 2 1
10. a.
11. a.
12. a. y
xO
11
y
xO
11
y
xO
1
1
b. 0.8, 5.2
c. 3 6 !5
b. 20.7, 2.7
c. 1 6 !3
b. 20.6, 23.4
c. 22 6 !2
b. 1.3, 4.7c. 3 6 !3
14580AK02.pgs 3/26/09 12:05 PM Page 291
13. a.
14. a.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26.
27. a.b. 0.2, 4.8
28. translated 6 right, 31 down29. translated 1 left, 3 down30. translated 3 right, 16 down31. translated left, 2 up32. reflected about x-axis, translated right, 2 or
up33. stretched vertically by a factor of 3, translated 1
unit left34. Vertex: (24, 211), axis of symmetry: x 5 24
Complete the square to get f(x) 5 (x 1 4)2 2 11.
Applying Skills
35. a. Width 5 ft, length 5 ftb.
5 5 38 ft2
c. Width 5 3.5 ft, length 5 10.9 ft36. a. Base 1 5 ft, base 2 5 ft,
height 5 ft
b.
5 5 20 ft2
c. Base 1 5 height 5 2.9 ft, base 2 5 10.9 ft37. Steve is 13, Alice is 15. Use x(x 1 2) 5 195.
A!6 2 1B A4!6 1 4BA A2!6 2 2B 1 A2!6B BQ2!6 2 2
2 R2!6 2 2
2!6 1 62!6 2 2
2A2!5 1 1B A2!5 2 1BA21 1 2!5B A2 1 4!5B
2 1 4!521 1 2!5
94
14
12
12
5 6 !212
23 6 !33221 6 !7
U12, 92V
3 6 !32
1 62!3
323 6 #152
4 6 2!33 6 !7
21 6 !62 6 !3
23 6 !51 6 !3
y
xO
1
1
y
xO1
1
292
5-2 The Quadratic Formula (pages 195–197)Writing About Mathematics
1. No. The denominator applies to all the terms inthe numerator.
2. Yes. When b2 , 4ac, the roots involve the squareroot of a negative number, which is not real.
Developing Skills3. 21, 24 4. 27, 1 5.
6. 7. 8.
9. 0, 3 10. 11.
12. 13. 14.
15. 16. 17.
18. a.
b. Answers will vary: 20.4, 25.6c.d. 20.4, 25.6
Applying Skills
19. or 20. Width 5 ft, length 5 ft21. Width 5 cm, length 5 cm22. Altitude 5 23 1 ft, base 5 3 1 ft23. Bases 5 8, 12; height 5 424. DB 5 22 1 , AD 5 2 1 , AB 5
25. a. and
b.
c.
d.
26. 11.9 seconds27. a. 1.2
b. 2.3c. 16.7
28.
The roots are the same.
x 52b 6 !b2 2 4ac
2a
Ax 1 b2a B 2
5 b2 2 4ac4a2
x2 1 bax 1 b2
4a2 5 b2
4a2 2 ca
2b2a
x 5 2b2a
A2b2a , 0 B
A2b 1 !b2 2 4ac2a , 0 BA2b 2 !b2 2 4ac
2a , 0 B4!372!372!37
3!53!52 1 !4622 1 !46
1 1 !321 1 !31 2 !6, 7 2 2!61 1 !6, 7 1 2!6
23 6 !7
y
xO
1
1
2 6 !103
12 6 !33 6 !6
1 6 !334
5 6 !334
1 6 !178
23, 121 6 !5
62!225 6 !332
1 6 !172
3 6 !52
b. 20.4, 2.4c. 1 6 !2
b. 2.4, 7.7c. 5 6 !7
14580AK02.pgs 3/26/09 12:05 PM Page 292
Hands-On Activity:Alternate Derivation of the Quadratic Formula
Yes.
5-3 The Discriminant (pages 201–203)Writing About Mathematics
1. a. 9
b.c. No. The rules apply only when a, b, and c are
rational numbers.2. Yes. Since b2 is always positive, when 24ac is
positive, b2 2 4ac . 0.Developing Skills
3. , 0 4. . 0 5. , 06. 5 0 7. . 0 8. 5 09. a. rational and unequal
b. 210. a. irrational and unequal
b. 211. a. rational and equal
b. 112. a. irrational and unequal
b. 213. a. not real numbers
b. 014. a. rational and unequal
b. 215. a. 0, rational and equal
b. 616. a. 49, rational and unequal
b.17. a. 5, irrational and unequal
b.18. a. 64, rational and unequal
b. 6219. a. 17, irrational and unequal
b.20. a. 211, not real numbers
b. no real roots21. a. 0, rational and equal
b.22. a. 49, rational and unequal
b.23. a. 211, irrational and unequal
b. no real roots24. Yes. A perfect square trinomial is the only way to
yield equal rational roots.
21, 52
12
1 6 !178
23 6 !52
0, 272
2!5 6 32
x 52b 6 "b2 2 4ac
2a
2ax 5 2b 6 "b2 2 4ac
2ax 1 b 5 6"b2 2 4ac
293
25. a. c 5 1b. any c , 1 such that 4 2 4c a perfect squarec. any c , 1 such that 4 2 4c is not a perfect
squared. c . 1
26. a. 64b. any b , 24 or b . 4 such that b2 2 16 is a
perfect squarec. any b , 24 or b . 4 such that b2 2 16 is not a
perfect squared. 24 , b , 4
Applying Skills
27. The fence cannot be constructed. Use. The discriminant is 2188,
so the equation has no real roots.28. Yes. Use . The discriminant is 0.29. Yes. Use . The determinant is
256.30. No. That value for the profit yields a negative
determinant.
5-4 The Complex Numbers (pages 208–209)
Hands-On ActivityFor the parallelogram with vertices 4 1 2i, 2 2 5i,and 0, the fourth vertex is 6 2 3i, which is the sum ofthe two given complex numbers.
In 1–9, the resulting complex number is always thesum of the two complex numbers. Student answersshould include graphs of parallelograms on thecomplex plane.
1. 5 1 5i 2. 27 1 7i 3. 6 2 4i4. 21 2 7i 5. 2 6. 2107. 23 2 4i 8. 3i 9. 4 1 2i
Writing About Mathematics1. No. Factoring out i from each term and then
multiplying yields the product 24.2. Yes. i ? i 5 i2 5 21 and any real coefficient, when
squared, is positive.Developing Skills
3. 2i 4. 9i5. 3i 6. 26i7. 211i 8.9. 10.
11. 12.13. 14.15. 16.17. 18. 23 1 6i19. 19i 20. 3i21. 13i 22. 3i23. 0 24. 7i!5
24 2 2i!61 1 i!35 1 i!510i!52i!5122i!515i!326i!22i!32i!2
216x2 1 48x 5 324x(5 2 x) 5 25
x2 1 (15 2 x)2 5 82
14580AK02.pgs 3/26/09 12:05 PM Page 293
25. 24 1 8i 26.27. 28.29. 30. 5 2 3i31. 23 1 7i 32.33. 34. 0.45 1 0.3i35. i 36. 2337. 6i 38. 25i39. i 40. 7 2 2i41. 1 42. 043. i 44. (2, 4)45. (22, 5) 46. (4, 22)
47. (24, 22) 48.
49. (0, 3) 50. (23, 0)51. (0, 0)Applying Skills
52. 13i ohms 53. 212i ohms
Hands-On Activity:Multiplying Complex NumbersMultiplication by i
1. 23 1 2i 2. 212
3. 4. 5
5. 210i 6. 2 2 3i
Multiplication by a real number
1. 8 1 16i 2.
3. 3 1 8i 4. 4i5. 22i 6. 6 2 9i
5-5 Operations With Complex Numbers(pages 215–216)
Writing About Mathematics1. Yes, i 2 5 21.2. Yes, (a 1 bi)(a 2 bi) 5 a2 2 b2i 2 5 a2 1 b2.
Developing Skills
3. 7 1 9i 4. 5 2 4i5. 9 2 4i 6. 24 2 2i7. 216i 8. 10i9. 7i 10. 22 2 19i
11. 0 12.
13. 14.
15. 16.
17. 18. 3 2 4i
19. 2 1 5i 20. 28 2 i
21. 26 1 9i 22.
23. 24.
25. p 2 2i 26. 29 1 7i
53 1 2
3i24 2 13i
12 1 3i
2542 2 13
24i
294 1 121
10 i2154 1 28
5 i
45 1 2
5 i83 2 7
6 i
34 2 1
4i
252 2 5
2 i
232 1 1
2 i
A 12, 4 B
21 1 i!6
37 2
i!24
21 2 4i!1024 1 i!72 1 6i!215!2 1 6i!2
294
27. 3 1 15i 28. 25 2 12i29. 2 2 23i 30. 11 1 23i31. 25i 32. 1733. 2148 34. 34i35. 1 36. 1
37. 21 38.
39. 40.
41. 42.
43. 44.
45. 46. 6 2 2i
47. 21 1 3i 48. 4 2 3i
49. 3 2 4i 50.
51. 52.
53. 54.
55. 1 2 4i 56. 3 2 4i
57. 58. 2i
59. 60.
Applying Skills
61. 62.
5-6 Complex Roots of a QuadraticEquation (page 219)
Writing About Mathematics1. Yes. b2 2 4ac will be negative, and since b 5 0,
there will be no real component.2. Yes. The two roots are made by adding and
subtracting the imaginary component from thesame real component.
Developing Skills
3. 2 6 2i 4. 23 6 i
5. 2 6 3i 6.
7. 25 6 2i 8. 24 6 i
9. 1 6 3i 10.
11. 12. 2 6 i
13. 14.
5-7 Sum and Product of the Roots of aQuadratic Equation (pages 223–224)
Writing About Mathematics1. x2 2 2px 1 p2 2 q 5 02. Both. Olivia’s equation is Adrien’s multiplied
by 2.Developing Skills
3. Sum 5 21 4. Sum 5 24Product 5 1 Product 5 5
1 6 i!22 6 i!3
212 6 2i
12 6 i
212 6 1
2 i
215 i4
3 1 6i
215 1 1
35 i64p2 1 1 1 12p
4p2 1 1 i
7125 1 1
125 i
2129 2 20
29 i45 1 3
5 i
75 2 11
5 i214 2 7
4 i
15 1 3
5 i
981 1 p2 2 p
81 1 p2 i
30349 2 108
349i817 1 2
17 i
85 1 4
5 i16 1 1
6 i
215 2 2
5 i110 2 1
5 i
12 2 1
2 i
14580AK02.pgs 3/26/09 12:05 PM Page 294
5. Sum 5 6. Sum 5
Product 5 21 Product 5 22
7. Sum 5 2 8. Sum 5 3Product 5 Product 5 21
9. Sum 5 8 10. Sum 5
Product 5 212 Product 5
11. Sum 5 12. Sum 5 0
Product 5 24 Product 5
13. Sum 5 0 14. Sum 5 22Product 5 1 Product 5 0
15. Sum 5 16. Sum 5 22
Product 5 Product 5 23
17. Sum 5 21Product 5
18. 210 19. 11 20.
21. 22. 23.
24. 1 25. 26. 3
27.
28. a.
b.c. The coefficients of the equation are not
rational numbers.29. a.
b.c. The coefficients of the equation are not
rational numbers.30. x2 2 7x 1 10 5 0 31. x2 2 11x 1 28 5 032. x2 2 x 2 12 5 0 33. x2 1 3x 1 2 5 034. x2 2 9 5 0 35. 4x2 2 16x 1 7 5 036. 32x2 2 12x 2 9 5 0 37. x2 2 x 5 038. x2 2 4x 1 1 5 0 39. x2 2 x 2 1 5 040. 9x2 1 6x 2 2 5 0 41. x2 2 6x 1 10 5 042. 4x2 2 12x 1 13 5 0 43. 4x2 1 9 5 0Applying Skills44. x2 2 15x 1 54 5 045. x2 2 12x 1 40 5 046. Answers will vary. Correct as long as .
Example: x2 2 4x 1 4 5 047. Sum:
Product:
5 b2 2 (b2 2 4ac)
4a2 5 ca
2b 1 "b2 2 4ac2a ? 2b 2 "b2 2 4ac
2a
22b2a 5 2b
a2b 1 "b2 2 4ac
2a 12b 2 "b2 2 4ac
2a 5
2b 5 c
3 1 11!3211 2 !3
6 2 9!2
3 2 !2
257
2114
5621
62134
252
53
298
234
214
52
294
12
43
225
32
295
48. c is the product of the roots. Since 2b is aninteger equal to the sum of the roots and oneroot is an integer, both roots are integers.Therefore, both roots are factors of c.
5-8 Solving Higher Degree PolynomialEquations (pages 227–228)
Writing About Mathematics1. Yes. This follows from the definition of a root.2. Yes. f(x 2 a) is a translation of f �a� units to the
right. Thus, each root is increased by �a�.Developing Skills
3. 0, 22, 25 4. 0, 1, 225. 23, 6 2i 6. 1,
7. 8. 3,9. 61, 62 10. 6i, 62i
11. 63, 63i 12.13. 61, 63 14. 0, 21, 2
15. 16.
17. 18. 22, 6119. a. 0 20. a. 216
b. Yes b. No21. a. 2 22. a. 0
b. No b. Yes23. a. 1 24. a. 0
b. No b. Yes25. a. 0 26. a.
b. Yes b. No27. a. 1 1 9i 28. a.
b. No b. NoApplying Skills29. a. Multiply out to check.
b. 1,c. Same as part b. The two equations are equal,
so they have equal roots.d. Prove by multiplying.
30. a. Multiply out to check.
b. 21,c. Same as part b. The two equations are equal,
so they have equal roots.d. Prove by multiplying.
31. a. The graph of g(x) is that of f(x) stretchedvertically by a factor of 2.
b. They are the same.c. The graph of p(x) will be that of q(x) stretched
vertically by a factor of a.d. They are the same.
Hands-On Activitya. 21, 1, 2 b. 22, 2, 3c. 22, 21, 3 d. 22, 21, 1, 2
12 6
!32 i
212 6
!32 i
2334 1 159
8 i
3 1 !3
61, 21 6 i!23
21, 12, 3 6 i!720, 63!2
612, 61
2 i
6!22 i3
2, 61
6i!3
14580AK02.pgs 3/26/09 12:05 PM Page 295
5-9 Solutions of Systems of Equations andInequalities (pages 236–239)
Writing About Mathematics1. The solutions of 0 . ax2 1 bx 1 c are the
x-coordinates of the solutions ofy . ax2 1 bx 1 c that are also on the x-axis.
2. The minimum value of x2 1 2 is (0, 2). The rangeis y $ 2 and never intersects y 5 22.
Developing Skills3. (21, 3) and (3, 3) 4. (24, 4) and (2, 22)5. (1, 5) and (3, 1) 6. (1, 4) and (4, 1)7. (2, 2) and (4, 4) 8. (1, 2) and (22, 21)9. (21, 2) and (4, 7) 10. (21, 21) and (1, 3)
11. (2, 2) and (3, 1) 12. (1, 4) and (4, 7)13. (20.5, 2.5) and (0, 3) 14. (0.5, 2.5) and (3, 5)15. (0.5, 2.25) and (4.5, 6.25) 16. (28, 26) and (3, 5)17. (3, 6) 18. (0, 0) and (5, 15)19. (21, 23) 20. (21, 2) and (22, 3)21. (2, 26) and (8, 6) 22. (0.5, 2) and (2, 5)23. (0.5, 3.25) and (4, 5) 24. (0.5, 4.5) and (3, 7)25. (22.5, 23.25) and (2, 21)26. (22.5, 30) and (3.5, 18)27. no real common solutions
28. and
29. and
30. and
31. and
32. (1.4, 15) and (0, 1)33. (4 1 2i, 4 2 2i) and (4 2 2i, 4 1 2i)34. (4, 2) and (2, 24)
35. and
36. y 5 22x, y 5 x2 2 237. y 5 2x 1 5, y 5 2(x 2 2)2 1 5
38. (1) 39. (4) 40. (2)
41. (3) 42. (6) 43. (5)
44. a. b. noy
xO21 1
Q4!55 , 8!5
5 RQ24!55 , 28!5
5 R
A2 1 !5, 28 1 11!5BA2 2 !5, 28 2 11!5BA1 1 !2, 5 1 2!2BA1 2 !2, 5 2 2!2B
A!3, 7 1 !3BA2!3, 7 2 !3BA!2, 6 1 4!2BA2!2, 6 2 4!2B
296
45. a. b. yes
46. a. b. no
47. a. b. yes
48. a. b. no
49. a. b. yes
y
xO 21 43
26272829210211212
2324 22
25
21
y
xO
1
1
y
xO 21 43
213214215216217218219
2223 21
y
xO 21 43 65 87 9
235236237238239240241242243
y
xO
11
14580AK02.pgs 3/26/09 12:05 PM Page 296
50. a. b. yes
51. a. b. no
Applying Skills52. Width 5 6 ft, length 5 8 ft53. 6 m, 7 m54. 4 ft by 8 ft55. a. (x 2 4)2 1 (y 2 2)2 5 20
b. Graphc. (0, 4) and (6, 22)
56. a. Graphb. Yes. The graphs intersect.c. and
57. a. Graphb. No. The graphs do not intersect.c. x 5 1 6 2i, y 5 2 6 4i
58. (22, 23), (0, 1), (2, 5)59. 0.4 , t , 2.860. a. x 2 4
b. V 5 2(x 2 4)2
c. x . 1261. a.
b. more than 20 and less than 1,060
y
x
1,400,000
100
1,300,0001,200,0001,100,0001,000,000
900,000800,000700,000600,000500,000400,000300,000200,000100,000
200
300
400
500
600
700
800
900
1,00
0
A3 1 !6, 8 1 2!6BA3 2 !6, 8 2 2!6B
y
xO
2
2
y
xO1
21
297
Review Exercises (pages 241–243)1. i 2. 4i 3. 3i
4. 5. 7i 6.7. 24i 8. 9. 7 2 i
10. 0 1 3i 11. 4 1 0i 12. 0 1 0i
13. 24 2 2i 14. 10 2 4i 15. 0 2 10i
16. 0 1 6i 17. 17 1 11i 18. 12 1 16i
19. 80 2 18i 20. 6 1 12i 21.22. 1 1 0i 23. 1 2 i 24. 2 1 0i
25. 26. 27. 215 1 8i
28. 5 2 12i 29. 30. 4, 23
31. 22 6 i 32. 33. 3 6 i34. 27.5, 4 35. 1 6 i 36. 0.5, 22
37. 38. 39.
40. 41. 42. 61, 62
43. 210 6 20i 44. 45.
46. 61, 63 47. 48. 26, , 1
49. translated 1 left, 1 up50. translated left, up51. scaled by 4, translated right, up52. reflected x-axis, scaled by 2, translated right,
down53. a. real, rational, unequal
b. real, irrational, unequalc. No. The parabola crosses the x-axis in two
distinct real points.54. Yes, the discriminant is positive.55. (2, 0) and (23, 5) 56. (1, 0) and (4, 3)57. (0, 0) and (5, 10) 58. (3, 4) and (4, 3)59. (6, 0) and (3, 3) 60. (21, 4) and (1, 2)61. (22, 6) and (3, 1) 62. (22, 24) and (4, 8)63. (21, 24) and (3, 0) 64. (3, 24) and (6, 12)
65. and
66. and
67. (20.5, 1.75) and
68. (1 2 i, i) and (1 1 i, 2i)69. (22 2 i, 25 2 2i) and (22 1 i, 25 1 2i)70. (2, 1) and (21, 22)71. x2 2 2x 2 15 5 072. or 2x2 1 7x 2 4 5 0
73. x2 2 5 74. x2 2 10x 1 7 5 075. x2 2 12x 1 40 5 0 76. 977. 24 , b , 4 78. c # 9
4
x2 1 72x 2 2 5 0
A 53, 5 B
Q21 1 !52 , 1 1 !5
2 RQ21 2 !52 , 1 2 !5
2 RQ7 1 !29
2 , 2 1 !29RQ7 2 !292 , 2 2 !29R
678
54
34
34
734
32
25324, 6!3
3
0, 3 6 !5223, 56
2, 6132 6 i
1 6 !65
1 6 !32
1 6 !52
3 6 !19
7 6 3!52
25 2 1
5 i2513 1 12
13 i
2 2 32i
4!63
7i!22i!3
14580AK02.pgs 3/26/09 12:05 PM Page 297
79. a. b. No
80. a. b. Yes
81. 8.3 m by 11.7 m
82. a.
b. $2 or $22c. The maximum profit is $100 when the price is
$12.83. The shorter side must be longer than 10 inches.
Exploration (pages 243–244)1.2. (x2 1 x 2 2)(x 2 2)3. 1 21 24 4 �—2
2 2 241 1 22 0
4. They are the same.
In (1)–(4), parts a and b, answers will vary dependingon the choice of root used.
1. c. Yesd. Roots 5 1, 2, 3; factors 5 (x 2 1), (x 2 2),
(x 2 3)
(x 2 1)(x 2 2)(x 1 2)
y
xO
102030405060708090
100
2 4 6 8 10 12 141618 2022
y
xO1
21
y
xO
11
298
2. c. Yesd. Roots 5 23, 21, 2; factors 5 (x 1 3), (x 1 1),
(x 2 2)3. c. Yes
d. Roots 5 1, 3; factors 5 (x 2 1), (x 2 1),(x 2 3)
4. c. Yesd. Roots 5 1, 2; factors 5 (x 2 1), (x 2 1),
(x 2 2)
Cumulative Review (pages 244–246)Part I
1. 3 2. 2 3. 24. 1 5. 2 6. 37. 3 8. 2 9. 1
10. 1
Part II
11.
12.
Part III
13.
14.
Part IV15. a–b.
16. a.b. h(x) 5 2x2 1 4
f + g(23) 5 2((23)2) 1 4 5 22
y
xO1
21
x 5 32x 5 23
2
2x 5 32x 5 23�2x� 5 3
3 2 �2x� 5 0
3 1 !53 2 !5 ? 3 1 !5
3 1 !5 514 1 6!5
4 57 1 3!5
2
x 5 2113x 2 9 5 4x 1 2
6 ? x 2 32 5 2x 1 1
3 ? 6
2 2 i3 2 i ? 3 1 i
3 1 i 5 710 2 1
10 i
14580AK02.pgs 3/26/09 12:05 PM Page 298
Chapter 6. Sequences and Series
6-1 Sequences (pages 250–252)Writing About Mathematics
1. Randi. Unless an upper limit is defined, thesequence is infinite.
2. a. Yes. an11 5 3(n 1 1) 2 1 5 3n 1 2 5(3n 2 1) 1 3 5 an 1 3.
b. Yes. an 5 2n for any integer n, including n 1 1.Developing Skills
3. 1, 2, 3, 4, 5 4. 6, 7, 8, 9, 105. 2, 4, 6, 8, 10 6.7. 8. 19, 18, 17, 16, 159. 3, 9, 27, 81, 243 10. 1, 4, 9, 16, 25
11. 5, 7, 9, 11, 13 12. 1, 3, 5, 7, 913. 14.
15. 21, 22, 23, 24, 25 16. 9, 6, 3, 0, 2317.
18.
19. a. an 5 2n 20. a. an 5 3nb. 18 b. 27
21. a. an 5 3n 2 2 22. a. an 5 3n
b. 25 b. 39 5 19,68323. a. an 5 24. a. an 5 2n 1 5
b. b. 23
25. a. an 5 12i 2 2ni 26. a. an 5
b. 26i b.27. a. an 5 28. a. an 5 n2 1 1
b. b. 8229. a. an 5 n ? (21)n11 30. a. an 5
b. 9 b. 331. 5, 6, 7, 8, 9 32. 1, 3, 9, 27, 8133. 1, 3, 7, 15, 31 34. 22, 4, 28, 16, 23235. 20, 16, 12, 8, 4 36. 4, 5, 7, 10, 1437. 38. 4, 10, 25, 62.5, 156.25
39.Applying Skills40. a. 30, 35, 40, 45, 50, 55, 60
b. an11 5 an 1 5, a1 5 3041. a. 4, 6, 8, 10, 12, 14, 16
b. an11 5 an 1 2, a1 5 442. a. 180, 178, 176, 174, 172, 170, 168, 166, 164
b. an11 5 an 2 2, a1 5 18043. a. Jan 1, Jan 8, Jan 15, Jan 22, Jan 29
b. an11 5 an 1 7, a1 5 144. a. $400, $440, $484, $532.40, $585.64, $644.20,
$708.62b. an11 5 1.1an, a1 5 $400
12, 2, 12, 2, 12
108, 36, 12, 4, 43
!n
910
nn 1 1
110
1n 1 1
2429 < 0.047
242n
12 1 i, 1 1 i, 32 1 i, 2 1 i, 52 1 i
43, 83, 4, 16
3 , 203
3, 2, 53, 32, 7512, 23, 34, 45, 56
12, 1, 32, 2, 52
1, 12, 13, 14, 15
299
45. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55Hands-On Activity
1. a. 3 b. 7 c. 15 d. 2n 2 12. an+1 5 2an 1 1, a1 5 3
6-2 Arithmetic Sequences (pages 256–257)Writing About Mathematics
1. Virginia’s solution works, but it is not a bettermethod. As the value of n increases, her methodbecomes more and more time-consuming.
2. No. Pedro’s method yields an arithmeticsequence of six terms, not five.
Developing Skills3. Yes, d 5 3 4. Yes, d 5 2i5. No 6. Yes, d 5 257. No 8. Yes, d 5 0.259. a. d 5 3 10. a. d 5 5
b. 24 b. 5711. a. d 5 22 12. a.
b. 0 b.13. a. d 5 22 14. a. d 5 0.1
b. 219 b. 4.015. 12, 18, 24, 30, 36, 4216. 120, 115, 110, 105, 100, 95, 90, 85, 8017. 6, 9, 12, 1518.19. an11 5 an 2 3Applying Skills20. $6,000, $5,600, $5,200, $4,800, . . .
The amount owed each month has a commondifference, 2400.
21. Week 9a. 60 5 20 1 (n 2 1)5b. 20, 25, 30, . . . , 60c. Using a formula is more efficient for long
sequences.22. a. Choose any linear function and set up a chart,
showing that for each integer value of x, yincreases by a fixed amount.
b. a1 5 b, d 5 m 23. a. 40
b. 154
6-3 Sigma Notation (pages 260–261)Writing About Mathematics
1. Yes. The first and last terms of the series havebeen decreased by 2, and then re-increased by 2in the expression evaluated by sigma.
73, 11
3
72
d 5 12
14580AK02.pgs 3/26/09 12:05 PM Page 299
2. is undefined for k 5 0.Developing Skills
3. a. 3 1 6 1 9 1 12 1 15 1 18 1 21 1 24 1 27 1 30b. 165
4. a. 0 1 2 1 4 1 6 1 8b. 20
5. a. 1 1 4 1 9 1 16b. 30
6. a. 1 1 8 1 27 1 64 1 125 1 216b. 441
7. a. 95 1 90 1 85 1 80 1 75 1 70 1 65 1 60 1 551 50
b. 7258. a. 12 1 15 1 18 1 21 1 24 1 27
b. 1179. a. (4 1 2i) 1 (9 1 2i) 1 (16 1 2i) 1 (25 1 2i)
b. 54 1 8i10. a. 21 1 2 2 3 1 4 2 5 1 6 2 7 1 8 2 9 1 10
b. 511. a. 14 1 17 1 20 1 23 1 26 1 29 1 32 1 35 1 38
1 41 1 44b. 319
12. a. 2i 2 2i 2 3i 2 4i 2 5i 2 6i 2 7i 2 8i 2 9i 2 10ib. 255i
13. a. 27 2 11 2 15b. 233
14. a. 0 1 4 2 64 1 1,296 2 32,768 1 1,000,000b. 968,468
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26.
Applying Skills27.28.
29. a. an 5 20 1 (n 2 1) ? 3 5 3n 1 17
b. a35
n51
f20 1 3(n 2 1)g 5 a35
n51
(3n 1 17)
5 (a1 1 a2 1 a3 1 c) 1 (b1 1 b2 1 b3 1 c)(a1 1 b1) 1 (a2 1 b2) 1 (a3 1 b3) 1 c
ka1 1 ka2 1 c 5 k(a1 1 a2 1 c)
a`
1A n
3n Ba`
n51
n2
a6
n51
1n(n 1 1)a
5
n51
(21)n A n3n B
a5
n51
1n!a
9
n51
nn 1 1
a5
n51
12n 2 1a
10
n51
3n
a19
n50
(100 2 5n)a5
n51
nn
a8
n51
(5n 2 4)a7
n51
(2n 1 1)
1k
300
30. a. an 5 45 1 (n2 1) ? 15 5 15n 1 30
b.
31. (1) 54.50(2) 12.74(3) 0.67
6-4 Arithmetic Series (pages 264–265)Writing About Mathematics
1. Yes. , yielding n 5 30. This is truefor any arithmetic series with 30 terms such that
.2. No. The difference between terms is not constant.
Developing Skills3. 42 4. 2105. 245 6. 60i7. 7 8.
9. a.
b. 210
10. a.
b. 72
11. a.
b. 60
12. a.
b. 322
13. a.
b.
14. a.
b. 290
15. a. b. 26
16. a.
b. 1,050
17. a.
b. 387.5
18. a.
b. 21619. a.
b. 1102 1 4 1 6 1 c 1 20
a12
n51
f7 1 2(n 2 1)g 5 a12
n51
(2n 1 5)
a10
n51
f27.5 1 2.5(n 2 1)g 5 a10
n51
(2.5n 1 25)
a20
n51
f100 2 5(n 2 1)g 5 a20
n51
f25n 1 105g
a12
n51
13n
a10
n51
22(n 2 1) 5 a10
n51
22n 1 2
1652
a15
n51
f2 1 12(n 2 1)g 5 a
15
n51
(12n 1 3
2)
a14
n51
f10 1 2(n 2 1)g 5 a14
n51
(2n 1 8)
a5
n51
f24 2 6(n 2 1)g 5 a5
n51
(26n 1 30)
a6
n51
f24 2 4.8(n 2 1)g 5 a6
n51
(24.8n 1 28.8)
a10
n51
f3 1 4(n 2 1)g 5 a10
n51
(4n 2 1)
120!2
a1 1 an 5 80
1,200 5 n2(80)
a5
n51
f45 1 15(n 2 1)g 5 a5
n51
(15n 1 30)
14580AK02.pgs 3/26/09 12:05 PM Page 300
20. a. b. 5621. a. b. 11022. a. b. 1,05023. a. b. 265024. a. b. 100Applying Skills25. 4526. a. 11 days b. 22 miles27. $120 28. 2,135 seats29. 375 minutes 30. $129,00031. $10,350
6-5 Geometric Sequences (pages 269–270)Writing About Mathematics
1. Answers will vary. This method works fine forsmall sequences, but is inefficient for large valuesof n.
2. Yes. There are three geometric means between 8and 32.
Developing Skills3. Yes, r 5 2 4. Yes, r 5 55. No, arithmetic 6. Yes, r 5 47. Yes, r 5 23 8.
9. 10. No, arithmetic11. Yes, r 5 210 12. Yes, r 5 0.113. Yes, r 5 22 14. Yes, r 5 a
15. 1; 6; 36; 216; 1,296 16.
17. 2, 6, 18, 54, 162 18.
19. 20. 10, 30, 90, 270, 81021. 21, 4, 216, 64, 225622. 100, 10, 1, 0.1, 0.01 or 100, 210, 1, 20.1, 0.0123. 1, 4, 16, 64, 256 or 1, 24, 16, 264, 25624. or 25. 1, 22, 4, 28, 1626. 81, 27, 9, 3, 1 or 81, 227, 9, 23, 127. 128
28.
29. 25630. 2,18731.32.
33. 15, 37.534. or
35. , 144, or , 144,Applying Skills36. $1,000, $1,060, $1,123.60, $1,191.02, $1,262.48,
$1,338.23, $1,418.52, $1,503.63, $1,593.85,$1,689.48
2432!2224!2432!224!2
24, 163 , 264
94, 163 , 64
9
13
6256!2
0.00032 or A 15 B 5
1, 2!2, 2, 22!2, 41, !2, 2, 2!2, 4
1, !2, 2, 2!2, 4
14, 21
2 , 1, 22, 4
40, 20, 10, 5, 52
Yes, r 5 13
Yes, r 5 13
1 1 3 1 5 1 c 1 1922 2 4 2 6 2 c 2 50100 1 95 1 90 1 c 1 520 1 18 1 16 1 c 1 25 1 6 1 7 1 c 1 11
301
37. $3,150, $3,307.50, $3,472.88, $3,646.5238. 5,000; 4,900; 4,802; 4,706; 4,611; 4,520; 4,429; 4,34139. 55, 61, 67, 73, 8140. $16,000, $12,800, $10,240, $8,19241. $42,500, $36,125, $30,706.25, $26,100.31,
$22,185.27, $18,857.48
6-6 Geometric Series (pages 272–273)Writing About Mathematics
1. Yes. an 5 a1rn21, so anr 5 a1r
n.2. Probably not. This method becomes especially
cumbersome with large values of n.Developing Skills
3. 4,095 4. 354,2925. 536,870,911.5 6. 1,111,1107. 409.5 8.
9. 1,275 10. 2,441,40611. 12. 6,55413. 39,364 or 19,684 14. 1,02315. a. 3 1 6 1 12 1 24 1 48 1 96
b. 18916. a.
b.
17. a.
b.
18. a. 26 2 24 2 96 2 384 2 1,536 2 6,144 2 24,5762 98,304 2 393,216
b. 524,28619. a. 1 2 2 1 4 2 8 1 16 2 32
b. 22120. a.
b.
21. a.
b.
22. a.
b.23.
24.
25. 1,02326. a. 400(1.05)1 5 $420
b. Yes, r 5 1.05c. $2,856.80
27. feet28. 20 days
1789
1 2 62587
1 2 62517
< 1,038.66
13 1 13!32547
9 5 260.7
281 1 27 1 29 1 c 1 219
3,17516 5 198.4375
100 1 50 1 25 1 c 1 10064
665243 < 2.7366
1 1 23 1 4
9 1 c
1 32243
31516 5 19.6875
10 1 5 1 52 1 c
1 56
364243 < 1.4979
1 1 13 1 1
9 1 c
1 1243
6364
310 2 12 ? 39 5
59,04839,366 < 1.49997
14580AK02.pgs 3/26/09 12:05 PM Page 301
6-7 Infinite Series (page 278)Writing About Mathematics
1.
2. No. The calculator’s value of e is only anapproximation. e is an irrational number.
Developing Skills
3. a.
b.
4. a.
b.
5. a.
b. Increases without limit
6. a.
b. Finite limit:
7. a.
b. Decreases without limit
8. a.
b. Finite limit: 12
9. a.
b. Finite limit: (e 2 2)
10. a.
b. Increases without limit
11. 12. 13.
14. 15. 16.
17. 1 # n , 25
Review Exercises (pages 280–281)1. a. an+1 5 an 1 4, a1 5 1
b. Arithmeticc. an 5 1 1 4(n 2 1) 5 4n 2 3d. 37
2. a. , a1 5 3b. Geometric
c.
d. 16,561
an 5 3 A 13 B n21
an11 5an3
126999
2499
1299
49
13
109
a`
n51
n(n 1 1)2
a`
n51
1(n 1 1)!
6 1 a`
n51
6 A 12 B n
5 1 a`
n51
(5 2 4n)
254
5 1 a`
n51
5 A 15 B n
a`
n51
2n
Finite limit: 83
2 1 a`
n51
2 A 14 B n
Finite limit: 32
1 1 a`
n51A 1
3 B n
5 cc 2 1
1
1 2 1c
S 5 a1 2 r 5
302
3. a. an11 5 an 2 1, a1 5 12b. Arithmeticc. an 5 12 2 1(n 2 1) 5 2n 1 14d. 3
4. a. an11 5 an 1 n 1 1, a1 5 1b. Neitherd. 55
5. a. an11 5 an 1 i(2n), a1 5 i
b. Neitherd. 1,023i
6. a. an11 5 23an, a1 5 2b. Geometricc.d. 239,366
7. 8.
9. 10.
11. 12.
13. 6, 11, 16, 21, 26 14. 15115. 71 16. 22017. 2, 5, 18.
19. 2,04820. an11 5 an 1 (6 1 2n), a1 5 1221. 1, 7, 31, 127, 51122. 6, 11, 16, 21, 26, 3123. 5, 25, 125 or 25, 25, 212524. 12
25.26. 3 1 9 1 27 1 81 5 12027. 12 1 9 1 6 1 3 1 0 2 3 2 6 5 2128. 60.26
29. a.
b. Finite limit: 630. a. 8 cans
b. 108 cans31. a. $24,500
b. $222,50032. a. $52,637.27
b. $368,569.0533. a. an 5 an–1 1 n 2 1
b. 0, 1, 3, 6, 10, 15, 21, 28, 36, 45
c. an 5n(n 2 1)
2
a`
n51
3 ? A 12 B n21
!22
59
28 < 7,629.3945252 , 125
4 , 6258
a`
n51
12na
7
n51
(21)n21 ? n
a6
n51
(2n 2 1)2a
6
n51
(2n 1 2)
a6
n50
(3n 1 2)a8
n51
(n)(n 1 1)2
an 5 2(23)n21
14580AK02.pgs 3/26/09 12:05 PM Page 302
Exploration (page 282)In 1–4, part a, answers will be graphs.
1. b. Diverges 2. b. Oscillates3. b. Converges 4. b. Converges
Cumulative Review (pages 283–285)Part I
1. 4 2. 2 3. 14. 1 5. 3 6. 37. 1 8. 3 9. 2
10. 4Part II
11. Answer:
12.
Part III
13.
14.�2 0 1 2 3 4 5�1
5 4 2 3i
5 3i 2 421
5 3 1 4ii ? ii
5 3 1 4ii
(2 1 i)2
i
x 5 7x 1 2 5 9
2!x 1 2 5 237 2 !x 1 2 5 4
Ax 1 32 B 2
5 94
x2 1 3x 1 94 5 9
4
94
303
Part IV
15. a.
b.
c.
Set both equations equal to each other:
Substitute this value of x into either equation tofind the y-coordinate.
16. a. an11 5 10an, a1 5 3
b.
c. 33,333
a5
n51
3(10)n21
x 5 212
28x 5 4x 2 1 5 9x 1 3
x 2 13 5 3x 1 1
A212, 21
2 B
y
xO1
21
y 5 x 2 13
Chapter 7. Exponential Functions
7-1 Laws of Exponents (pages 288–289)Writing About Mathematics
1. No, they do not share a common base or commonexponent. (2)3(5)2 5 (8)(25) 5 200. 105 5 10,000.
2. Yes, this is true via the commutative property.Developing Skills
3. x7 4. y6 5. x4
6. y3 7. x10 8. 8y12
9. 106 10. 228 11. x6y3
12. x2y7 13. 29x6 14. 9x6
15. x5y 16. x 17. x8y10
18. 64x10 19. 16 20. x5y5
21. xy2 22. x2y2 23.
24. 32a5b 25. 4abc4 26. b
y7z2
x2
Applying Skills27. 9 28. 329. 30. x 5 25y2
31. $608.33 32. $3,909.3533. 15 years
7-2 Zero and Negative Exponents (pages 292–293)
Writing About Mathematics1. No. a0 1 a0 5 2a0 5 2(1) 5 22. Yes. a0 1 a0 5 1 1 1 5 2 and 2a0 5 2.
Developing Skills3. 4. 5.
6. 2 7. 125 8. 32
136
116
15
y 5 x3
14580AK02.pgs 3/26/09 12:05 PM Page 303
9. 16 10. 11. 112. 1 13. 1 14. 2115. 1 16. 4 17. 2218. 1 19. 1 20.21. 1 22. 3 23. 124. 25. 26.
27. 28.
29. � 74.8309 30.
31. 32. 33.
34. 56 35. 36.
37. 38. 39.
40. 41. 42.
43. 44. 45.
46. 47. x3 48. y7
49. 3a5 50. 6a4 51. 9a3x2
52. 53. 54.
55. 56. 57.
58. 59. 60. 9b
61. 62. 63.
64. y22 65. ab22 66. y24
67. 6b21 68. a3 69. 3x24
70. 2a23 71. 4x5 72. 3a73. 5xy210 74. 5x23y23 75. 25a6b24
76. 77. 78. a 5 8b 5 3
79.
7-3 Fractional Exponents (pages 296–298)Writing About Mathematics
1.
2.Developing Skills
3. 2 4. 3 5. 106. 2 7. 5 8. 69. 2 10. 6 11. 2
12. 15 13. 240 14. 34315. 32 16. 81 17. 1,00018. 16 19. 20.
21. 22. 23. 9
24. 5 25. 49 26. 8
125
11,000
12
13
Aa12B1
2 5 a12?1
2 5 a14 5 !4 a
A!n a B0 5 a0n 5 a0 5 1
3 3 1100
4921
8
x4 1 1x5
uv3
7232b4
xa2
x9
y21
2ab4x3y4
8m9
27n6b15c12
a15z2
xy5
2y2
a421
x2
14x2
2 181a42 1
4x212
181a4
14x2
25y8
7a4
2x2
1y5
1a6
1x4
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36. 1.196 37. 38.
39. 40. 41.42. 43. 44.
45. 46. 47. 7x
48. 49. 50.
51. 52. 3a 53.
54. 55. 56. 5
57. 58.59. 60.
61. 62.
63. 64.
65. 66.
67. 68.
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71. 5 72. 5
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78. 79.
80. 81. 5
82.
83. a.
✔
b.
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3 5?
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A!3 27 ? !3 3 B2 5? A2713 ? 3
13B2
232xy
54
27
15
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15x7
15y1415
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1a
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14580AK03.pgs 3/27/09 10:53 AM Page 304
7-4 Exponential Functions and TheirGraphs (pages 302–303)
Writing About Mathematics1. Any non-zero number raised to the zero power
is 1.2. One raised to any power is 1, thus y 5 1 for all
values of x.Developing SkillsIn 3–6, parts a and b, answers will be graphs.
3. c. or
4. c. or
5. c. or 6. c. or
7. a.
b. 3.1 c. 4.4
8. a–b.
c. y 5 22x
9. a–b.
c. y 5 21.2x
y
xO11
y
xO
1
1
y
O 1
1
2
3
2x
4
A 43 B xA 3
4 B2xA 27 B xA 7
2 B2x
y 5 A 13 B x
y 5 32x
y 5 A 14 B x
y 5 42x
305
10. a.
b.
c. 1.6Applying Skills11. a.
b. In 2010, 338,880,723.In 2020, 386,313,106.
y
xO 1 2 3 4 5 6 7 8 9 1011 121314
450400350300250200150100
50
y
O 1
1
2
3
x
4
21
x ex
–2 0.135
–1 0.368
0 1
1 2.718
2 7.389
3 20.086
14580AK03.pgs 3/27/09 10:53 AM Page 305
12. a.
b. In 10 years, 9.997 grams.In 100 years, 9.971 grams.
c. Answers will vary: 79,951 years13. a. Graph
b. 2 points: (2, 16) and (4, 64)c. y 5 4x
7-5 Solving Equations Involving Exponents(page 305)
Writing About Mathematics1. Yes. Squaring both sides eliminates the fractional
exponent.2. No. a22 5 , but 36 does not equal its inverse.
Developing Skills3. 64 4. 32 5. 2436. 4 7. 8. 29. 10. 81 11. 16
12. 9 13. 27 14. 7215. 3 16. 5 17. 8118. 0.35 19. 14.70 20. 1.2421. 2.03 22. 2.20 23. 0.5424.
Applying Skills
25. If the area of one face is B, then the length of oneside of the cube is . Therefore, the volume ofthe cube is or .
26.
7-6 Solving Exponential Equations (pages 307–308)
Writing About Mathematics1. a 5 0. Anything to the zero power is 1.2. There is no common base.
B 5 V23
B32A!BB3
!B
x 5 11,000
Ax213B23 5 (10)23
x213 5 10
x13 22
3 5 10
16
613
1a2
y
x
O
10,0
00987654321
20,0
0030
,000
40,0
0050
,000
60,0
0070
,000
80,0
0090
,000
306
Developing Skills3. 32 4. 33 5. 52
6. 72 7. 103 8. 25
9. 10. or 11. (0.1)3
12. (0.5)3 13. (0.9)2 14. (0.4)2
15. 4 16. 3 17. 2118. 22 19. 2 20. 2221. 22. 2 23. 3
24. 21 25. 26. 22
27. 3 28. 3 29. 630. 3 31. 2 32. 2233. 1 34. 35.36. 0 37. 23 38. 62
7-7 Applications of Exponential Functions(pages 312–313)
Writing About Mathematics1. 100% 5 1; thus, A 5 A0(1 1 1)n 5 A0(2)n.2. Daily interest earns interest on earned interest,
not just the principal.Developing Skills
3. 7.39 4. 4.48 5. 0.376. 2.72 7. 0.23 8. 2.729. 2,980.96 10. 168.50 11. 51.01
12. 344.60 13. r 5 100% 14. t 5 315. 577.21% 16. 236.11%Applying Skills17. a. $1,060, $1,123.60, $1,191.02, $1,262.48,
$1,338.23b. $1,061.68, $1,127.16, $1,196.68, $1,270.49,
$1,348.85c. Sued. Joe 5 6%, Sue 5 6.168%
18. a. $10,129.08b. $10,272.17
19. $1,508,661.82 20. 17.22 g21. 31,529 22. $369,452.8023. 30.23 g 24. 4,00025. a. A 5 A0e
rt; medicine decreases continually.b. Continuous 5 69.99 mg
Periodic 5 66.16 mg
Review Exercises (pages 315–316)1. 1 2. 3.
4. 5 5. 64 6. 500
7. 36 8. 9. 3
10. 11. 25 12.
13. 2 14. 15. 16 116
128
164
110
136
12
12
32
15
12
212
623A 16 B 3A 1
2 B 3 or 223
14580AK03.pgs 3/27/09 10:53 AM Page 306
16. 10,000 17. 59,049x2 18.
19. 20. 21.
22. 23.
24.
25. 26.
27. 28.
29. a–b.
c. rotation about the y-axis
30. 31. 64 32.33. 34. 35.
36. 22 37. 2 38. 439. 21 40. 41. 2142. $520.30 43. 9.05 mg 44. 4.7%45. $6,553.60 46. (0.5, 0.346)
Exploration (page 316)
a. 5
b.
Cumulative Review (pages 316–318)Part I
1. 2 2. 2 3. 34. 2 5. 1 6. 27. 2 8. 4 9. 1
10. 4
Part II11. The common difference d is 2.
The first term a1 is 1.
an 5 1 1 (n 2 1)2 5 2n 2 1
5 4(4a 1 1)4a12 1 42 5
4(4a 1 1)42(4a 1 1) 5
14
4a11 1 42a15 ? 2a21 1 16 5
4(4a 1 1)22a14 1 16
235 8
12
8943
3a 2 3a22
3a21 1 3a 53a(1 2 322)3a(321 1 1)
5
12
32
12
12
94
16
y
xO1
21
ab"4 a2b2cA !4 a 1 2B3
8
4x!4 y4y2!2y
3216x
43y
12 or 2
56x
43y
12
bA64112a
712b
23B or 2
12a
712b
53 3
y23
35x
12
6yz12
x1
x35b
25
c6
a6 or A ca B 6
307
12.
or
Part III
13.
14.
Part IV
15. Answer:Use the quadratic formula to find the roots of thecorresponding equation:
Test a number from each interval formed by theroots to find the solution.
16. a. (x 2 2)2 1 y2 5 16b. x2 1 y2 2 4x 2 12 5 0c. y 5 x 1 2d. Answer: (2, 4), (22, 0)
Substitute y 5 x 1 2 into the equation of thecircle:
Substitute into the equation of theline to find the y-coordinates.
x 5 62
x 5 62
2x2 5 8
2x2 2 8 5 0
x2 1 x2 1 4x 1 4 2 4x 2 12 5 0
x2 1 (x 1 2)2 2 4x 2 12 5 0
5 7 6 !972 < 21.42, 8.42
5 7 6 !49 1 482
x 52(27) 6 "(27)2 2 4(1)(212)
2(1)
7 2 !972 , x ,
7 1 !972
x 5 13
33x 5 3133x13 5 3427x11 5 81
1 1 27x11 5 82
5 215 1 7
5i
5 3 1 4i 2 45
3 1 2i1 2 2i ? 1 1 2i
1 1 2i
2x2 2 9x 1 10 5 0
x2 2 92x 1 5 5 0
x2 2 52x 2 2x 1 5 5 0
(x 2 2) Ax 2 52 B 5 0
14580AK03.pgs 3/27/09 10:53 AM Page 307
Chapter 8. Logarithmic Functions
308
7. a.b.
8. a.b.
9. a.b.
10. a. f21(x) 5 log2 (2x)
b.
11. y 5 log6 x 12. y 5 log10 x13. y 5 log8 x 14. y 5 log0.1 x15. y 5 log0.2 x 16.17. 18. y 5 2x
19. y 5 5x 20. y 5 10x
21. y 5 8x 22. y 5 (0.1)x or y 5 10–x
y 5 log 112 x
y 5 log14 x
y
xO1
1
y
xO1
1
f21(x) 5 log13 x
y
xO21
1
f21(x) 5 log!2 x
y
xO1
1
f21(x) 5 log0.5 x8-1 Inverse of an Exponential Function(page 323)
Writing About Mathematics1. Yes. The point (0, 1) is on the graph of any
exponential function y 5 bx. Therefore, since y 5 logb x is the inverse of the exponentialfunction y 5 bx, (1, 0) is always on its graph.
2. Yes. x 5 b2y, thus 2y 5 logb x, .Developing Skills
3. a. f21(x) 5 log3 x
b.
4. a. f21(x) 5 log5 x
b.
5. a. f21(x) 5 log1.5 x
b.
6. a. f21(x) 5 log2.5 x
b. y
xO1
1
y
xO21
1
y
xO1
21
y
xO121
y 5 12 logb x
14580AK03.pgs 3/27/09 10:53 AM Page 308
Applying Skills23. a. (0, 1), (1, 3), (2, 9), (3, 27)
b. (1, 0), (3, 1), (9, 2), (27, 3)24. a. (0, 1), (1, 1.05), (2, 1.10), (3, 1.16)
b. (1, 0), (1.05, 1) (1.10, 2), (1.16, 3)
8-2 Logarithmic Form of an ExponentialEquation (pages 326–327)
Writing About Mathematics
1. , so if , then .
2. If , then .
Developing Skills3. log2 16 5 4 4. log5 125 5 35. log8 64 5 2 6. log12 1 5 07. log6 216 5 3 8. log10 0.1 5 219. log5 0.008 5 23 10. log4 0.0625 5 22
11. 12.
13. 14.15. 102 5 100 16. 53 5 12517. 42 5 16 18. 27 5 12819. 35 5 243 20. 70 5 121. 10–3 5 0.001 22. 100–1 5 0.0123. 5–2 5 0.04 24.25. 26.27. 1 28. 5 29. 330. 12 31. –1 32. 2833. –2 34. 26 35. 636. 2 37. –3 38.39. 4 40. 4 41. 1642. 3 43. 3 44. 545. 2 46. 2 47. 248. 2 49. 23 50. –451. 90 52. 8 53. 854. 2 55. 56.
57. 1,000 58. 5 59. 460. 2 61. 3 62. 263. 64. 5 0.00000256
65. 66. 67.
68. 100 69. 4 70.
71. 23 72. 73.
74. 4
Applying Skills75. t 5 log1.06 A 76. n 5 log0.97 R77. a. loge A or ln A
b. t 5 2,500 loge A or t 5 2,500 ln A
110
35
12
122!21
10
1390,625
125
59
1615
2163
32225 5 0.2549
32 5 343
813 5 2
log100 0.001 5 232log625 125 5 3
4
log64 4 5 13log7
17 5 21
b2a 5 c2ba 5 c
b2a 5 1cba 5 cb2a 5 1
ba
309
8-3 Logarithmic Relationships (pages 331–332)
Writing About Mathematics1. by the logarithm of
a power rule and the logarithm of the base rule.2. No. For example, and
.Developing Skills
3. log3 (27 3 81) 5 log3 27 1 log3 81 5 3 1 4 5 7� 27 3 81 5 37 5 2,187
4. log3 (243 3 27) 5 log3 243 1 log3 27 5 5 1 3 5 8� 243 3 27 5 38 5 6,561
5. log3 (19,683 4 729) 5 log3 19,683 2 log3 729 5 9 2 6 5 3� 19,683 4 729 5 33 5 27
6. log3 (6,561 4 27) 5 log3 6,5612 log3 27 5 8 2 3 5 5� 6,561 4 27 5 35 5 243
7. log3 94 5 4 log3 9 5 4 3 2 5 8� 94 5 38 5 6,561
8. log3 2432 5 2 log3 243 5 2 3 5 5 10� 2432 5 310 5 59,049
9. 2 log3 81 1 log3 9 5 2 3 4 1 2 5 10� 812 3 9 5 310 5 59,049
10.
�
11.
�
12.
� 5 31 5 3
13.
� 813 4 5 39 5 19,683
14.
�
15. a. log3 9 16. a. log3 2,187b. 2 b. 7
17. a. 18. a. log3 27b. 21 b. 3
19. a. log3 27 20. a. log3 3b. 3 b. 1
21. a. 4(log3 9 2 log3 27) 22. a.b. 24 b. 3
23. a. log4 16b. 2
24. loge 10x 25. log2 ab
26. log2 (x 1 2)4 27. log10 y
(y 2 1)2
12(log3 3 1 log3 243)
log3 13
27 3 #3 72919,683 5 32 5 9
5 3 1 13(6 2 9) 5 2
log3 27 1 13(log3 729 2 log3 19,683)
!729
3 log3 81 2 12 log3 729 5 12 2 3 5 9
!19,683 4 2,187
12(log3 19,683 2 log3 2,187) 5 1
2(9 2 7) 5 1
!4 243 3 2,187 5 33 5 27
14(log3 243 1 log3 2,187) 5 1
4(5 1 7) 5 3
!6,561 4 729 5 322 5 19
12 log3 6,561 2 log3 729 5 4 2 6 5 22
(log10 10) ? (log10 10) 5 1log10 10 ? 10 5 2
loga an 5 nloga a 5 n ? 1 5 n
14580AK03.pgs 3/27/09 10:53 AM Page 309
28. 29. log3 x3
30. log2 2 1 log2 a 1 log2 b 5 1 1 log2 a 1 log2 b
31. log3 10 2 log3 x 32. 25 log5 a
33. 2 log10 (x 1 1) 34. 6 log4 x 2 5 log4 y
35. 36. A 1 B
37. 2A 1 B 38. 3(A 1 3)39. A 1 3B 40. A 2 B
41. 2A 2 3B 42.
43. 44.
45. 46.
47. 48. 32 49. 32
50. 32 51. 25 52. 253. 10
8-4 Common Logarithms (pages 335–336)Writing About Mathematics
1. log 80 5 log (10 3 8) 5 log 10 1 log 8 5 1 1 log 82. 10 must be raised to a negative power to yield
values less than 1.Developing Skills
3. 0.57 4. 0.935. 1.68 6. 1.757. 2.75 8. 3.759. 20.47 10. 21.12
11. 0 12. 113. 2 14. 2115. 3.01 16. 1.9017. 22.70 18. 1.4319. 0.60 20. 79.5921. 2.58 22. 1.5723. 21.49 24. 3.790525. 6.7562 26. 24.132427. 60.1174 28. 159.587929. 364.6700 30. 66,069.344831. 0.2902 32. 0.876433. 0.0701 34. 0.001035. 0.0001 36. x 1 y37. 2x 38. 2y39. 2x 1 y 40. x 1 2y41. 3x 42. 2x43. 2y 44. 22y45. 2x 2 1 46. x 2 y47. 2(x 2 y) 48. 2c49. 1 1 c 50. 2 1 c51. c 2 1 52. 2 2 c53. 2c 2 1 54. 2c 2 255. 1
2c
14B
32AA 2 B
2
12A 2 3BA 1 1
2B
12(A 1 B)
12 loge x
loge x ? y2
z2
310
56.
57.
Applying Skills58. a.
b.
c. Double in the 15th year, triple in the 24th year59. a. 7.40 b. 2.19 c. 4.40
8-5 Natural Logarithms (pages 338–339)Writing About Mathematics
1. The bases of the logarithms do not affect theanswer. If and , then
.2. a 5 1. logb 1 5 0 for any positive b � 1.
Developing Skills3. 1.32 4. 2.15 5. 3.876. 4.03 7. 6.33 8. 8.639. 21.07 10. 22.58 11. 0
12. 1 13. 2 14. 2115. 20.69 16. 3.56 17. 0.5518. 13.82 19. 0.39 20. 0.3521. 1.7837 22. 2.2926 23. 3.985224. 5.9239 25. 9.0521 26. 12.960427. 123.9651 28. 0.5843 29. 0.944330. 0.3152 31. 0.3679 32. 0.135333. x 1 y 34. 2y 35. 2x36. 2x 1 y 37. 3x 1 y 38. 2x 1 2y39. 2y 40. 2x 41. 2x 2 y42. 22(x 1 y) 43. x 2 y 44. 2(x 2 y)45. 2c 46. 3c 47. 2c48. 2c 49. 22c 50. 22c51. 52. 53. 3.55554. 5.380 55. 0.693 56. { }
57.
58.
5 2 1 ln x 1 12 ln y 2 ln z
2 ln e 1 ln x 1 12 ln y 2 ln z
ln !x ? z3
y
12c1
2c
ay 5 bz 5 xlogb x 5 zloga x 5 y
t
KO 5
10
1015 20 25 30
203040506070
12 log x 1 1
2 log y 2 12 log z
log82 ? (x2 2 4)
6 5 log32(x2 2 4)
3
K 1 2 3 4 5
t 0 15.403 24.414 30.807 35.765
K 10 20 30
t 51.169 66.572 75.582
14580AK03.pgs 3/27/09 10:53 AM Page 310
Hands-On Activity:The Change of Base Formulaa. 2.11 b. 4.09 c. 1.16d. 1.84 e. 20.73 f. 22.58g. 4.95 h. 20.39
8-6 Exponential Equations (pages 343–344)Writing About Mathematics
1. No. You must take the logarithm of each side, noteach term. This equation can be solved by firstsubtracting 6 from each side and then taking thelog of both sides.
2. No. The exponent is applied only to 3, not to theentire left side.
Developing Skills3. 2.26 4. 4.17 5. 2.866. 1.70 7. 2.23 8. 4.089. 6.86 10. 2.38 11. 20.5
12. 20.15 13. 2.14 14. 2.89Applying Skills15. 15 years old 16. 5 years17. 25.5 years 18. 25 years19. a. 66 minutes 20. a. 20.00251
b. 22 minutes b. 828 days
8-7 Logarithmic Equations (page 346)Writing About Mathematics
1. No. The left side must first be combined using therules for logarithms: log x 1 log 12 5 log 12x.Thus, the equation can be solved by writing 12x 5 9.
2. Yes. Taking the logarithm of a number equal to the base is equivalent to 1. Then log x 5 log (10 ? 5).
Developing Skills3. 25 4. 6 5. 1.56. 30 7. 126 8. 1929. 4 10. 9 11. 3
12. 4 13. 5 14. 0.515. 500 16. 1 17. {1.38, 3.62}18. 3.65
Review Exercises (pages 348–350)1. a. Graph b. {x : x . 0}
c. {all real numbers} d. Graphe. y 5 3x
2. y 5 6x 3. y 5 2.5x
4. 5. log2 8 5 3
6. log6 36 5 2 7. log10 0.1 5 218. 9.
10. 11. 34 5 81log2 14 5 22
log8 4 5 23log3!3 5 1
2
y 5log xlog 82 or y 5 log82 x
311
12. 53 5 125 13.
14. 15. 1021 5 0.116. e0 5 1 17. 918. 19. 21220. 21. 322. 22 23.24. 4 25. a 1 b26. 2a 27. a 1 2b28. 29.
30. 2(b 2 a) 31.32. 33. 134. 2.5 35.
36. 9 37.
38. 23 39.40.
41. a.b. 22
42. a. log360 (5 3 12 3 6) 5 log360 360b. 1
43. a.b. 4
44. a. 5
b. 245. a. 2 ln 42 2 ln 3
b. 6.3846. a. 2 ln 14 1 ln 0.625
b. 4.8147. a. 4 ln 0.25 2 ln 26 1 5 ln 3
b. 23.3148. A 5 xy 49.
50. 51. A 5 x ? y3
52. 53.
54. 3.5 55. 3 56. 0.7557. 3 58. 2 59. 1.2460. 2,013 61. 16 62. 4,500
Exploration (pages 350–351)Steps 1–8. an gn
0 1
1 10
1.125 13.3352143
1.25 17.7827941
1.5 31.6227766
2 100
A 5 x ? y13 or A 5 x!3 yA 5 A x
y B 2
A 5yx
A 5 xy
log1.5 94log1.5 A 3
2 3 3 3 12 B
log0.5 16
256 5 log0.5 116
log4 !4 8148
14
!5
136
52
2a 2 32b
13(a 2 b)
12b 2 a1
2(a 1 2b)
52
214
569
712 5 !7
432 5 8
14580AK03.pgs 3/27/09 10:53 AM Page 311
Step 9.Step 10. The mean of 1.25 and 1.5 is 1.375. Thus,
log (23.71373706) � 1.375.
Cumulative Review (pages 351–352)Part 1
1. 4 2. 2 3. 44. 4 5. 4 6. 27. 4 8. 3 9. 2
10. 3Part II
11.
x 5 0
12. Answer: x2 2 10x 1 34 5 0, 5 6 3iLet a 5 1.
r1r2 5 c 5 34r1 1 r2 5 2b 5 10
x 5 612
4x2 5 1x(4x2 2 1) 5 0
f(x) 5 4x3 2 x 5 0
log (13.3352143) < 1.125
312
Part III
13.
14.
Part IV15. a.
b. 37.49990416.
t < 3,900 years
t 5 ln 15 2 ln 9.2520.000124
ln 9.25 5 ln 15 2 0.000124 t9.25 5 15e20.000124 t
30 1 6 1 1.2 1 0.24 1 c 1 0.000384
5 19!25 10!2 1 5!2 1 4!2
!200 1 !50 1 2!8
x 5 1log 5 < 1.43
3x log 5 5 3log 53x 5 log 1,000
53x 5 1,000
Chapter 9.Trigonometric Functions
9-1 Trigonometry of the Right Triangle(pages 356–357)
Writing About Mathematics1. They are equal. By definition, if A is an angle
on a right triangle, then and.
2. Yes. Since lengths are positive values and thelength of a leg of a right triangle is always smallerthan the length of the hypotenuse, sin A is apositive value less than 1.
Developing Skills
3. a. b. c.
4. a. b. c.
5. a. b. c.
6. a. b. c.
7. a. b. c.
8. a. b. c.
9. a. b. c.
10. a. b. c.
11. The triangles are similar and therefore have thesame trig ratios.
2!55
!53
23
!147
!73
!23
12
2!55
!55
815
1517
817
815
1517
817
1160
6061
1161
512
1213
513
34
45
35
cos (90 2 A) 5 ah
sin A 5 ah
12. sin 45° 5 cos 45° 5 , tan 45° 5 1
13. sin 45° 5 , cos 45° 5 , tan 45° 5
Applying Skills
14. sin 5 , cos 5 , tan 5
15. 0.25
16. sin 5 , cos 5 , tan 5
17. 15 m 18. 125 ft 19. 56 ft
9-2 Angles and Arcs as Rotations (pages 360–361)
Writing About Mathematics1. Yes, 810° 5 90° 1 (2)360°.2. No, two angles that add to 360 are not necessarily
coterminal. For example, 150° and 210°.Developing SkillsIn 3–7, answers will be graphs.
3. In quadrant I 4. Same as 180°5. Same as 180° 6. Same as 240°7. In quadrant II 8. I9. II 10. III 11. IV
12. IV 13. II 14. I15. IV 16. I 17. IV18. 30° 19. 52° 20. 280°21. 350° 22. 275° 23. 90°
512
1213
513
43
35
45
!33
!32
12
!22
14580AK03.pgs 3/27/09 10:53 AM Page 312
24. 220° 25. 180° 26. 0°27. 260°Applying Skills28. Clockwise 29. Counterclockwise30. a. Clockwise b. 2,340°31. 60 32. 12.533. a. 87 s 34. a. 18° per second
b. 6 min, 15 sec b. 540° per secondc. 3,600° per second
9-3 The Unit Circle, Sine, and Cosine (page 366)
Writing About Mathematics1. Since P is a point on the unit circle, the largest
value for either x or y is 1 and the smallest valueis 21.
2. No. For example, sin 45° 5 sin 135°.Developing Skills
3. a. b. c. I
4. a. 20.8 b. 0.6 c. IV
5. a. b. c. II
6. a. b. c. IV
7. a. b. c. III
8. a. b. c. IV
9. a. b. c.
10. a. b. c. II
11. 90° 12. 270°13. 0° 14. 180°15. (0, 1) 16. (21, 0)17. (0.2, 1.0) 18. (20.7, 0.7)19. (20.7, 20.8) 20. (0.7, 20.7)21. (21, 0) 22. (20.7, 0.7)
23. a.
b.
c.
Applying Skills24. a. (5 cos u, 5 sin u)
b. (25 cos u, 25 sin u)c. m�ROP9 5 u, m�ROP0 5 180 1 u
25. a. (2cos u, sin u)b. u
c. 180 2 u
23
6!53
6!53
2 941
4041
I!22
!22
24252 7
25
2 513212
13
!552
2!55
2!32
12
35
45
313
Hands-On Activity3. Answers will vary: (0.94, 0.34)5. The values are about the same.6. Answers will vary: 70° (0.34, 0.94);
100° (20.17, 0.98); 165° (20.97, 0.26);200° (20.94, 20.34); 250° (20.34, 20.94);300° (0.50, 20.87); 345° (0.97, 20.26)
In each case, the values of the sine and cosine areapproximately equal to the coordinates of P.
Hands-On Activity: Finding Sine and Cosine UsingAny Point on the Plane
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
9-4 The Tangent Function (pages 372–373)Writing About Mathematics
1. a. 45° and 225°. If P is a point on the unit circleand on a 45° angle in standard position, anisosceles right triangle is formed by the x- and y-coordinates of P. Thus, the x- and y-coordinates of P are equal and the sine andcosine of 45° are equal. A similar result holdsfor 225° by symmetry.
b. 45° and 225°. Since , if tan u 5 1,then sin u 5 cos u.
2. cos u 5 0. Since , tan u is undefinedwhen the denominator is zero.
Developing Skills
3. a. b. c. d.
4. a. b. c. d.
5. a. b. c. d.
6. a. b. c. d.
7. a. b. c. d. 2!33
122
!32
12
!152!15
42142
!154
243
4523
5810
2125212
135
1321213
43
45
35
45
sin u cos u 5 tan u
sin u cos u 5 tan u
r 5 3!2, sin u 5 2!22 , cos u 5 2!2
2
r 5 !5, sin u 522!5
5 , cos u 5!55
r 5 5!2, sin u 57!2
10 , cos u 52!2
10
r 5 5!2, sin u 57!2
10 , cos u 5 !210
r 5 !53, sin u 527!53
53 , cos u 522!53
53
r 5 17, sin u 5 21517 , cos u 5 8
17
r 5 13, sin u 5 1213, cos u 5 25
13
r 5 5, sin u 5 45, cos u 5 3
5
14580AK03.pgs 3/27/09 10:53 AM Page 313
8. a. b. c. d.
9. a. b. c. d. 1
10. a. b. c. d.
11. a. b. c. d.
12. a. b. c. d.
13. a. 25 b. c. d.
14. a. 17 b. c. d.
15. a. b. c. d. 1
16. a. 5 b. c. d.
17. a. b. c. d. 22
18. a. b. c. d. 23
19. a. b. c. d. 21
20. a. b. c. d.
21. I 22. IV 23. III24. I 25. II 26. III27. a. 0
b. Undefinedc. Answers will vary: 270° 1 360°n
28. a. 0b. 61c. Answers will vary: any multiple of 180°
29. a. 61b. 0c. Answers will vary: 90° 1 180°n
Applying Skills30. a. {u : u is any degree angle}
b. {y : 21 # y # 1}c. No, tan u is undefined at 90° 1 180°n.d. {u : u � 90° 1 180°n}e. {all real numbers}
31. Apply the Pythagorean Theorem with x 5 cos uand y 5 sin u.
32.
Hands-On Activity3. Answers will vary: 0.345. The values are about the same.6. Answers will vary: 70° (1, 2.75); 100° (1, 25.67);
165° (1, 20.27); 200° (1, 0.36); 250° (1, 2.75);300° (1, 21.73); 345° (1, 20.27)In each case, the value of the tangent isapproximately equal to the y-coordinate of P.
m 5yx 5 sin u
cos u 5 tan u
2132
!1010
3!10103!10
2!22
!224!2
3!10102
!10102!10
2!552
!553!5
4324
5235
2!222
!22!2
158
1517
817
247
2425
725
2!14
7!232
!73
!23
!73
!74
34
!74
22!622!6
5152
2!65
2!222
!222
!22
2!55
23
!53
23
314
9-5 The Reciprocal TrigonometricFunctions (pages 377–378)
Writing About Mathematics
1. Since , for sec u to equal one-half,cos u has to equal 2, which is not possible.
2.
Developing Skills3. a. 0.8 b. 0.6
c. 0.75 d.
e. 1.25 f.
4. a. 20.28 b. 0.96
c. d.
e. f.
5. a. b.
c. d. 26
e. f.
6. a. b.
c. d. 22
e. f.
7. a. b.
c. d.
e. 3 f.
8. a. b.
c. d.
e. f.
9. a. b.
c. d.
e. f.
10. a. b.
c. d.
e. f.
11. a. 5 b.
c. d.
12. a. b.
c. d. 14!17
!174!17
34
53
54
2!10327
3
27!10
203!10
20
22!10
7237
2!14
65!2
6
25!7
723!14
7
2!75
3!25
!5223
2
23!5
52!5
5
2!5322
3
2!2
3!24
!24
2!23
13
2!332
2!33
2!3
212
!32
2!3535
6!3535
2!35
216
!356
2247 < 23.4286225
7 < 23.5714
2524 5 1.04162 7
24 5 20.2916
43 5 1.3
53 5 1.6
cot 908 5 1 tan 908 5 cos 908
sin 908 5 01 5 0
sec u 5 1 cos u
14580AK03.pgs 3/27/09 10:53 AM Page 314
13. a. b.
c. d. 1
14. a. b.
c. d. 1
15. a. b.
c. d. 21
16. a. b.
c. d.
17. a. b.c. d. 21
18. a. b.
c. d. 23
19. a. 1, 21 b. 0 c. 1, 2120. a. 1, 21 b. 0 c. 1, 2121. a. 0 b. 1, 21 c. 022. 1, 21 23. 90°Applying Skills24. 7.5 feet25. a. 26. a. 12 mi
b. b. 7.5 mi
c. c. 12 mi
27. a. Divide the given equation by cos2 u.b. No. It is true only where tan u and sec u are
defined.c. OT
28.
9-6 Function Values of Special Angles(pages 380–381)
Writing About Mathematics
1. Yes. If P9 5 (x, y), then and .2. Yes.
Developing Skills
3. 4. 5. 2
6. 7. 8.
9. 2 10. 11.
12. 13. 14. !22
!33!3
2!33
!32
12!3!3
3
12
!32
sin u 5 6"1 2 a2
sin2 u 1 a2 5 1 sin2 u 1 cos2 u 5 1
sin u 5yr cos u 5 x
r
5 cos u sin u , sin u 2 01
sin u cos u
cot u 5 1 tan u 5
512
1312
135
!103
2!103!10
!22!29!2
2122!5
!524!5
2!2
!26!2
2!2
2!25!2
2!2
2!23!2
315
15. 16. 17.
18. 1 19. 1 20. 2121. 21 22. 0 23. Undefined24. 0 25. Undefined 26. 027. Undefined 28. 21 29. 2130. Undefined 31. 0 32. 1
33. 1 34. 35. 3
36. 1 37. 1 38.
39. 1 40. 41. 1
42. 43. 1 44.
Applying Skills
45.46. Answers will vary:
✘
47. Answers will vary. Example:
9-7 Function Values from the Calculator(pages 384–385)
Writing About Mathematics1. tan 90° is undefined.2. 400° and 40° are co-terminal angles.
Developing Skills3. 0.4695 4. 0.8192 5. 4.70466. 20.1736 7. 0.1736 8. 0.36409. 0.3640 10. 20.2588 11. 0.2588
12. 20.9848 13. 0.9848 14. 0.267915. 0.2679 16. 20.5736 17. 20.573618. 20.1736 19. 20.1736 20. 20.173621. 0.9500 22. 0.8450 23. 38.188524. 0.9621 25. 20.1352 26. 20.904827. 20.4258 28. 16.1190 29. 3.236130. 3.8637 31. 0.5095 32. 25.758833. 1.2208 34. 23.7321 35. 21.103436. 0.2867 37. 1.6616 38. 24.445439. 20° 40. 64° 41. 12°42. 40° 43. 35° 44. 87°45. 3° 46. 62° 47. 33°
cos (308) 5!32 . cos (608) 5 1
2
308 , 608
1 2!32
12 1 1
2 5!32
sin (308) 1 sin (308) 5? sin (608)
120 ft 3 120!3 ft
14
2!33
12
14
!2 1 12
!2!22!2
14580AK03.pgs 3/27/09 10:53 AM Page 315
48. 57° 49. 46° 50. 85°51. 15° 309 52. 74° 309 53. 75° 459
54. 14° 159 55. 40° 489 56. 49° 129
57. 82° 159 58. 5° 069
Applying Skills59. 17° 60. 20° 299
61. 51° 099, 47° 609, 80° 609
62. a. II 63. a. IVb. 143° b. 286°
9-8 Reference Angles and the Calculator(page 391)
Writing About Mathematics1. Yes, 2u is equivalent to 360 2 u.2. No. Only sin and tan are negative in quadrant IV.
Cos will return a positive value.
Developing Skills
3. a–c. d. 60°
4. a–c. d. 70°y
x70°
70°250°
y
x60° 60°120°
316
5. a–c. d. 40°
6. a–c. d. 45°
7. a–c. d. 45°
8. 80° 9. 5° 10. 30°11. 70° 12. 75° 13. 50°14. 85° 15. 70° 16. 50°17. 35° 18. 2sin 35° 19. 2cos 85°20. tan 75° 21. cos 48° 22. 2tan 10°23. 2sin 75° 24. 2cos 65° 25. 2tan 55°26. 2sin 56° 27. sin 40° 28. 20°, 160°29. 51°, 309° 30. 18°, 198° 31. 55°, 235°32. 54°, 126° 33. 183°, 357° 34. 108°, 252°35. 23°, 337° 36. 96°, 276° 37. 14°, 166°38. 138°, 222° 39. 188°, 352° 40. 172°, 352°41. 0°, 180° 42. 90°, 270° 43. 0°, 180°
y
x
405°
45°
y
x
45°
45°
245°
y
x
40°
40°320°
14580AK03.pgs 3/27/09 10:53 AM Page 316
Review Exercises (pages 394–396)1. a–c. d. 40°
2. a–c. d. 60°
3. a–c. d. 35°
4. a–c. d. 80°y
x
80°
80°
2100°
y
x35° 35°
145°
y
x
60°
60°300°
y
x40°
40°220°
317
5. a–c. d. 60°
6. a. 145°b. 58°
7. a. IV 8. a. I
b. 20.6 b.
c. 0.8 c.
d. 20.75 d.
e. e.
f. 1.25 f.
g. g.
9. a. II 10. a. III
b. b.
c. c.
d. d.
e. 5 e.
f. f.
g. g.
11. a. I 12. a. Quadrantal
b. b. 0
c. c. 21
d. d. 0
e. e. Undefined
f. f. 21
g. g. Undefined43
54
53
34
45
35
2!5522!6
2322
5!612
23!5
5
!522
!612
2232
2!65
2!53
15
3!7724
3 5 21.3
43
4!7725
3 5 21.6
!73
34
!74
y
x
60°
60°600°
14580AK03.pgs 3/27/09 10:53 AM Page 317
13. a. III 14. a. IV
b. b.
c. c.
d. d.
e. 2 e.
f. 2 f.
g. g.
15. a. (0.7, 0.7) b. (0.2, 1.0)
16. 17. 18. 1
19. 20. 21.
22. 23. 24.
25. 2 26. 27. 22
28. 0.6428 29. 5.6713 30. 0.939731. 1.1918 32. 20.9925 33. 20.406734. 20.6428 35. 0.1763 36. 2cos 80°37. 2sin 60° 38. cos 80° 39. tan 30°40. sin 30° 41. 2cos 75° 42. cos 40°43. tan 50° 44. 22°, 158° 45. 24°, 336°46. 54°, 234° 47. 224°, 316° 48. 150°, 330°49. 145°, 215° 50. 44°, 316° 51. 19°, 161°52. 90°, 270° 53. 0°, 180° 54. 90°, 270°55. 0°, 180°56. a. q b. p c. t
d. By similar triangles, . In parts a–c, weshowed that , , and
. It follows that bysubstitution.
57. 6° 509 58. 18 feet
Exploration (page 396)In �OTR:
Since , m�u 5 m�QSO. Thus:
cot u 5 cot /QSO 5adjopp 5
QSQO 5
QS1 5 QS
csc u 5 csc /QSO 5hypopp 5
OSOQ 5
OS1 5 OS
QS || OR
sec u 51
cos u 5hypadj 5
OTOR 5
OT1 5 OT
tan u 5 sin u cos ut 5 tan u
p 5 cos uq 5 sin u
qp 5
t1
!33
!2!32
!32
2!322
!222!3
!32
!32
2 913
43
5!109
54
25!10
1353
2139
34
9!105024
5
213!10
50235
318
Cumulative Review (pages 396–398)Part I
1. 1 2. 3 3. 24. 4 5. 2 6. 47. 1 8. 3 9. 1
10. 4Part II11.
12.
Part III13. Answer: x 5 6
Check x 5 1: Check x 5 6:
✘ ✔
14. Since
r 5 ,
the equation of the circle is:
(x 2 2)2 1 (y 2 1)2 5 5Part IV
15. Answer:Write the equation in standard form and thenuse the quadratic formula with a 5 1, b 5 26,and c 5 4:
16. a.
b.
5 25x2 1 65x 1 425 25x2 1 70x 1 49 2 5x 2 75 (5x 1 7)2 2 5x 2 7
f + g(x) 5 f(5x 1 7)5 (23)2 1 3 5 12
f + g(22) 5 f(5(22) 1 7) 5 f(23)
x 56 6 !20
2 56 6 2!5
2 5 3 6 !5
x 56 6 !36 2 16
2
x 52(26) 6 !(26)2 2 4(1)(4)
2(1)
x 5 3 6 !5
"(4 2 2)2 1 (1 2 0)2 5 "22 1 12 5 !5
6 5 65 2 13 1 (9)
12 5? 63 1 (4)
12 5? 1
3 1 (6 1 3)12 5? 63 1 (1 1 3)
12 5? 1
x 5 1, 60 5 x2 2 7x 1 6
x 1 3 5 x2 2 6x 1 9(x 1 3)
12 5 x 2 3
3 1 (x 1 3)12 5 x
12 , x , 7
2
1 , 2x , 723 , 2x 2 4 , 3
�2x 2 4� , 35 21 1 5i
(3 2 2i)(21 1 i) 5 23 1 3i 1 2i 1 2
14580AK03.pgs 3/27/09 10:53 AM Page 318
Chapter 10. More Trigonometric Functions
319
25. a. 200° b.c.
26. a. 270° b.c.
27. a. 500° b.c.
28. 6 29. 2 30. 2531. 3.2 32. 6 33. 4034. 5 35. 4p 36.37. 6p 38. 3.4 in. 39. 2.440. 1.5 m
15p
y
x
40°140°
2p9
y
x270°
7p18
7p18
y
xp9
200°
p910-1 Radian Measure (pages 404–406)
Writing About Mathematics1. Yes, when the length of the intercepted arc is
divided by the radius of the circle, the unitscancel, giving equivalent ratios.
2. 4p. Two full revolutions is 720° 5 4p radians.Developing Skills
3. 4. 5.
6. 7. 8.
9. 10. 11.
12. 13. 60° 14. 20°
15. 18° 16. 72° 17. 200°18. 270° 19. 540° 20. 330°21. 630° 22. 57.3°23. a. 60° b.
c.
24. a. 35° b.c. y
x35° 7p
36( )
7p36
y
x
60° p3( )
p3
11p6
3p2
4p3
5p4
3p4
8p9
2p3
p4
p2
p6
14580AK04.pgs 3/26/09 12:06 PM Page 319
41.
Applying Skills42. 7.2 ft 43. 944. a. Yes, one complete revolution for any circle is
2p radians.b. No. The radian measure is the same but the
length of the radius is not, so the measure ofthe arc, and therefore the distance traveled,will differ accordingly.
45. 2,457.4 km
10-2 Trigonometric Function Values andRadian Measure (pages 409–410)
Writing About Mathematics1. Yes, p 5 180° so the formula is correct.2. Yes. Adding any multiple of 2p 5 360° keeps the
terminal side the same.Developing Skills
3. 4. 5. 0
6. 7. 8.
9. 1 10. 2 11. Undefined12. 1 13. 0.2771 14. 1.002915. 1.3086 16. 0.3514 17. 0.973218. 0.5976 19. 0.9912 20. 0.579621. 1.3785 22. 1.2542 23. 0.440424. 1.3750 25. 26. 0
27. 28.Applying Skills29. (20.4236, 20.9059)30. a. 2.50 b. 2.50
c. (20.8011, 0.5985) d. (22.4034, 1.7954)e. Since for both points, x is negative and y is
positive, P and B are both in quadrant II.31. a.
b. 20.161532. 28.0 ft33. a. 2,405 ft
b. 2,352 ftHands-On Activity 1:The Unit Circle and Radian Measure
1. (cos 0, sin 0) 5 (1, 0);
;
;
;
A cos p2 , sin
p2 B 5 (0, 1)
A cos p3 , sin
p3 B 5 A 1
2, !32 B
A cos p4 , sin
p4 B 5 A !2
2 , !22 B
A cos p6 , sin
p6 B 5 A !3
2 , 12 B
16623
25!3
61 2!22
12
2!3221
2!33
!3!22
320
2. ;
;
;
;
;
;
;
;
;
;
Hands-On Activity 2:Evaluating the Sine and Cosine Functions
1.
2. 0.70713. 0.5000
10-3 Pythagorean Identities (pages 413–414)
Writing About Mathematics
1. Yes.
2. Yes. Both equations are equivalent to the identity.
Developing Skills
3. , , ,
, ,
4. , , , ,
,
5. , , ,
, ,
6. , , ,
, ,
7. , , ,
, , csc u 5 32 sec u 5 2
3!55cot u 5 2
!52
tan u 5 22!5
5 cos u 5 2!53sin u 5 2
3
csc u 5 232 sec u 5
3!55cot u 5 2
!52
tan u 5 22!5
5 cos u 5!53sin u 5 22
3
csc u 5 24!7
7 sec u 5 243cot u 5
3!77
tan u 5!73 cos u 5 23
4sin u 5 2!74
csc u 54!7
7sec u 5 43
cot u 53!7
7 tan u 5!73 cos u 5 3
4sin u 5!74
csc u 5 5 sec u 5 25!6
12cot u 5 22!6
tan u 5 2!612 cos u 5 2
2!65sin u 5 1
5
cos 2 u 1 sin
2 u 5 1
( sec u)(csc u) 5 1 cos u ? 1
sin u 5 1 cos u sin u
cos x: x8
8!, 2x10
10!
sin x: x9
9!, 2x11
11!
A cos 11p
6 , sin 11p
6 B 5 Q!32 , 21
2RA cos
7p4 , sin
7p4 B 5 Q!2
2 , 2!22 R
A cos 5p3 , sin
5p3 B 5 Q1
2, 2!32 R
A cos 3p2 , sin
3p2 B 5 (0, 21)
A cos 2p3 , sin
2p3 B 5 Q21
2, 2!32 R
A cos 5p4 , sin
5p4 B 5 Q2!2
2 , 2!22 R
A cos 7p6 , sin
7p6 B 5 Q2!3
2 , 212R
( cos p, sin p) 5 (21, 0)
A cos 5p6 , sin
5p6 B 5 Q2!3
2 , 12RA cos
3p4 , sin
3p4 B 5 Q2!2
2 , !22 R
A cos 2p3 , sin
2p3 B 5 Q21
2, !32 R30° 45° 60° 90° 180° 270° 360°
2p3p2pp
2p3
p4
p6
14580AK04.pgs 3/26/09 12:06 PM Page 320
8. , , ,
, ,
9. , , ,
, ,
10. , , ,
, ,
11. , , ,
, ,
12. , , , ,
,
13. , , , ,
,
14. , , ,
, ,
15. 1 16. cos u 17. sin u
18. 19. 20.
21. 1 22. 0
10-4 Domain and Range of TrigonometricFunctions (page 419)
Writing About Mathematics
1. Yes, . Both functionsare undefined for integer multiples of p.
2. No, cot .
Developing Skills3. 1 4. 05. Tangent is undefined at (n 5 0).6. Secant is undefined at (n 5 0).7. 1 8. 0 9. 0
10. Cotangent is undefined at np (n 5 1).11. Secant is undefined at (n 5 1).12. 21 13. 014. Cotangent is undefined at np (n 5 0).15. Tangent is undefined at (n 5 21).16. 2117. Secant is undefined at (n 5 5).18. Cotangent is undefined at np (n 5 28).19. Answers will vary:20. Answers will vary: np
p2 1 np
p2 1 np
p2 1 np
p2 1 np
p2 1 np
p2 1 np
5 01 5 0
cos p2
sin p2
p2 5
cot u 5 cos u sin u 5 cos u ? csc u
sin u cos u
1 1 sin u cos u
1 sin u
csc u 5 2!37 sec u 5!37
6cot u 5 26
tan u 5 216 cos u 5
6!3737sin u 5 2
!3737
csc u 5 54 sec u 5 25
3
cot u 5 34 tan u 5 24
3 cos u 5 235 sin u 5 4
5
csc u 5 54 sec u 5 25
3
cot u 5 234 tan u 5 24
3 cos u 5 235 sin u 5 4
5
csc u 5 2!34
3 sec u 5 2!34
5cot u 5 53
tan u 5 35 cos u 5 2
5!3434sin u 5 2
3!3434
csc u 58!7
21 sec u 5 28cot u 5 2!721
tan u 5 23!7 cos u 5 218sin u 5
3!78
csc u 5 2!17
4 sec u 5 2!17cot u 5 14
tan u 5 4 cos u 5 2!1717sin u 5 2
4!1717
csc u 5!52 sec u 5 2!5cot u 5 21
2
tan u 5 22 cos u 5 2!55sin u 5
2!55
321
10-5 Inverse Trigonometric Functions(pages 423–425)
Writing About Mathematics1. No. The restricted domain of cosine is 0 # x # p,
while the restricted domain of tangent is . These two intervals are not
equivalent.2. Yes. The calculator returns an equivalent answer
for cos21 (20.5) regardless of whether it is indegrees or radians.
Developing Skills3. a. 30° 4. a. 45° 5. a. 0°
b. b. b. 06. a. 245° 7. a. 120° 8. a. 230°
b. b. b.9. a. 260° 10. a. 260° 11. a. 180°
b. b. b. p
12. a. 90° 13. a. 0° 14. a. 90°b. b. 0 b.
15. 37° 16. 127° 17. 77°18. 277° 19. 26° 20. 154°21. 46° 22. 246° 23. 287°
24. 25. 1 26. 0
27. 21 28. 29.
30. 21 31. 32.
33. 34. 35.
36. p 37. 38.
39. a.
b.
c.
Applying Skills
40. a.
b. No. The restricted domain of secant does notinclude .
c. (2`, 21] � [1, `)d. (2`, 21] � [1, `)
e.
41. a.
b. No. The restricted domain of cosecant doesnot include 0.
c. (2`, 21] � [1, `)d. (2`, 1] � [1, `)
e.
42. a.
b. No. The restricted domain of the tangentfunction does not include 0.
A2p2 , 0 B < A0, p2 B
S2p2 , 0) < A0, p2 T
S2p2 , 0 B < A0, p2 T
S0, p2 B < Ap2 , pT
p2
S0, p2 B < Ap2 , pT
u 5 arctan A x 1 1x 1 2 B
u 5 arcsin A 32x 1 3 B
u 5 arccos A xx 1 1 B
p62p
4
p4
p6
p4
!22
!22
12
2!33
!22
p2
p2
2p32p
3
2p6
2p32p
4
p4
p6
2p2 , x , p
2
14580AK04.pgs 3/26/09 12:06 PM Page 321
c. (2`, `)d. (2`, `)e.
43. a.
b. 63°
10-6 Cofunctions (pages 427–428)Writing About Mathematics
1. Yes. Cofunctions allow you to express anytrigonometric function in terms of the sinefunction. Also, reference angles allow you toexpress any trigonometric function value interms of an acute angle.
2. No. If A is in quadrant II, .Developing Skills
3. a. cos 25° 4. a. sin 10°b. 0.9063 b. 0.1736
5. a. cot 36° 6. a. cos 4°b. 1.3764 b. 0.9976
7. a. sec 42° 8. a. csc 15°b. 1.3456 b. 3.8637
9. a. tan 33° 10. a. sin 20°b. 0.6494 b. 0.3420
11. a. cos (220°) or cos 20° 12. a. cot (25°)b. 0.9397 b. 211.4301
13. a. sin (240°) 14. a. csc (235°)b. 20.6428 b. 21.7434
15. a. cos (2140°) 16. a. sin (2165°)b. 20.7660 b. 20.2588
17. a. cot (2147°) 18. a. sec (2176°)b. 1.5399 b. 1.0024
19. a. sin (2210°) or sin 30° 20. a. cos (2205°)b. 0.5 b. 20.9063
21. a. tan (2222°)b. 20.9004
22. a. csc (2195°) or csc 15°b. 3.8637
23. 35° 24. 20°25.
26. a. 27. a.
b. b.
28. a. 29. a.
b. b. 22!33
csc A2p6 Bcot
p3
!22
!32
sin p4 cos
p6
cos A 52!3
2
u 5 arctan 1d
A2p2 , 0 B < A 0, p2 B
322
30. a. 31. a.
b. 2 b. Undefined
32. a. 33. a.
b. b.
34. a.
b.
Review Exercises (pages 430–431)1. 2.
3. 4.
5. 45° 6. 72°7. 210° 8. 222.5°9. 10. u 5 1
11. s 5 10 cm 12. u 5 1.513. r 5 4 cm 14. u 5 3015. r 5 4 cm 16. r 5 2.5 cm17. s 5 2p ft18. a. 90°, 270° b. 0°, 180°, 360°
c. 90°, 270° d. 0°, 180°, 360°
19. a. Domain: [21, 1], Range:
b. Domain: [21, 1], Range:c. Domain: {all real numbers}, Range:
20. 21. 22.
23. 1 24. 25. 21
26. 27. 28. 22
29. 2 30. 0 31. 2132. 2 33. 12 34.
35. 36. Undefined 37.
38. 39.
Exploration (page 431)1.
2OT 5 sec u:Let T 5 (0, 2t).
y
x
S Q
R
T
P
O
122!2
!22!33
2!32
13
2!3221
2
212
!3!22
12
A2p2 , p2 B
f0, pg
S2p2 , p2 T
p2
2p3
5p4
3p4
5p12
!3
cot 13p
6
2122
!22
sin A213p6 B cos
3p4
tan A2p2 B sec A2p
3 B
csc u 5 sec Ap2 2 u B sec u 5 csc Ap
2 2 u Bcot u 5 tan Ap
2 2 u B tan u 5 cot Ap2 2 u B
sin u 5 cos Ap2 2 u B cos u 5 sin Ap
2 2 u B
14580AK04.pgs 3/26/09 12:06 PM Page 322
Let T9 5 the image of T about a reflection in thex-axis.
Then OT 5 OT9 and �ROT9 5 180 2 u is a first-quadrant angle.From the Chapter 9 Exploration,
.Using the properties of reference angles,
.OS 5 csc u:Let S9 5 the image of S about a reflection in the
y-axis.Then OS 5 OS9 and �ROS9 5 180 2 u is a first-quadrant angle.From the Chapter 9 Exploration,
Using the properties of reference angles,
.2QS 5 cot u:Let S9 5 the image of S about a reflection in the
y-axis.Then QS 5 QS9 and �ROS9 5 180 2 u is a first-quadrant angle.From the Chapter 9 Exploration,
Using the properties of reference angles,
.A similar procedure can be used to prove steps 2and 3.
2.
3. y
x
S Q
R
TP
O
y
x
SQ
R
T
P
O
cot u 5 2cot (180 2 u) 5 2QS
cot (180 2 u) 5 QSr 5 QS
csc u 5 csc (180 2 u) 5 OS
csc (180 2 u) 5 OSr 5 OS
sec u 5 2 sec (180 2 u) 5 2OT
sec (180 2 u) 5 OTr 5 OT
323
Cumulative Review (pages 431–433)Part I
1. 3 2. 4 3. 24. 3 5. 4 6. 37. 4 8. 4 9. 2
10. 3Part II
11.
12.
Part III13. Answer: (x 2 2)2 1 (y 2 2)2 5 80
The radius of the circle
The center of the circle
14. a.
b. 1Part IV
15.
16.
(1, 22), (4, 1)
y
xO
1
21
5 2!2
tan u 5 2" sec2 u 2 1 5 2"A!3B2 2 1
sin u 5 2"1 2 cos2 u 5 2#1 2 A !33 B 2
5 2!63
cos u 5 1 sec u 5 1
!3 5!33
5 log13A 729
9 ? 243 B 5 log13 13 ? 1
243
19
A 127 B
2log13
5 (2, 2)
5 A22 1 62 , 4 1 0
2 B5 4!5
5 "82 1 42
5 "(22 2 6)2 1 (4 2 0)2
2808 ? p180 5 14p
9
x 5 3 6 2i
x 56 6 !216
2
x 56 6 !36 2 4(1)(13)
2(1)
14580AK04.pgs 3/26/09 12:06 PM Page 323
Chapter 11. Graphs of Trigonometric Functions
324
c.
d. 24.15 ft
11. a.b. 2p # x # p
c.
Y3 always gives the better approximation.
11-2 Graph of the Cosine Function (pages 445–447)
Writing About Mathematics1. Yes. For every (x, y) on the graph there is also a
point (2x, y) on the graph.2. Yes. The period of y 5 sin x is 2p.
Developing Skills3. Graph
a. p , x , 2p, 3p , x , 4p
b. 0 , x , p, 2p , x , 3p
c. 2 cycles4. 15. 216. 2p
7. No. It fails the horizontal line test.
2p2 # x # p
2
25 sin 5p12 <
h
u5p12
p12
25
6
11-1 Graph of the Sine Function (pages 440–441)
Writing About Mathematics1. Yes, since for each (x, y) on the graph there is
also a point (2x, 2y) on the graph.2. Yes. The period of y 5 sin x is 2p.
Developing Skills3. Graph
a. , ,
b. ,
c. 2 cycles4. 1 5. 21 6. 2p
7. No. It fails the horizontal line test.8. a.
b. 1 2np or 1 2np
Applying Skills9. a–b.
2u is the reflection of u about the x-axis.sin u 5 2sin(2u)
c. Yes. For all angles in the four quadrants,sin u 5 2sin(2u).
d. Yes. sin 0° 5 2sin (20°) 5 sin 180°5 2sin (2180°) 5 0
sin 90° 5 2sin (290°) 5 sin 270°5 2sin (2270°) 5 1
e. Yes, since for all x in the domain,f(x) 5 2f(2x).
10. a.
b. h(u) 5 25 sin u
p12 # u # 5p
12
y
x
R
P
O 2u
P9
u
1
2p3
p3
cos p3 5 1
2
5p2 , x , 7p
2p2 , x , 3p
2
7p2 , x # 4p3p
2 , x , 5p20 # x , p
2
x sin x Y2(x) Y3(x)
0.5 0.50000213 0.49999999
0.70714305 0.70710647
p 0 0.52404391 20.0752206
!22 < 0.70710678p
4
p6
14580AK04.pgs 3/26/09 12:06 PM Page 324
Applying Skills8. a–b.
2u is the reflection of u about the x-axis.cos u 5 cos (2u)
c. Yes. For all angles in the four quadrants,cos u 5 cos (2u).
d. Yes. cos 0° 5 cos (20°) 5 cos 180°5 cos (2180°) 5 1
cos 90° 5 cos (290°) 5 cos 270°5 cos (2270°) 5 0
e. Yes. For all x in the domain, f(x) 5 f(2x).9. a. d(u) 5 6 cos u
b.
c.
d. 5.901 feet
10. a.
b.
c.
Y3 always gives the better approximation.
23p4 # x # 3p
4
2p2 # x # p
2
6 cos p18 <
d
u
p36
5.98
6
p18
5.96
5.94
5.92
5.9
p36 # u # p
18
y
x
R
P
O 2u
P9
u
1
325
11-3 Amplitude, Period, and Phase Shift(pages 453–455)
Writing About Mathematics1. Yes. The first graph is shifted to the left and the
second is shifted to the right, resulting in thegraphs starting p units apart. Since this is equalto the period of each curve, their graphs willcompletely overlap.
2. No. If we factor out a 2 from the secondequation, we see that its graph is shifted units,in contrast to a shift of units for the first graph.
Developing Skills3. 1 4. 2 5. 5
6. 3 7. 8.
9. 0.6 10. 11. 2p
12. 2p 13. 14. p
15. 4p 16. 6p 17.
18. 19. 20.
21. 22. 23.
24. 25. 2p 26.
27.
28. y
xO
p 2p
1
21
y
xO
p 2p
1
21
p223p
4
p6
p42p
3
p22p
28p3
4p3
2p3
18
12
34
p4
p8
p2
p2
x cos x Y2(x) Y3(x)
0.86605388 0.86602526
0.70742921 0.70710321
2p 21 0.12390993 21.211353
!22 < 0.707106782p
4
!32 < 0.866025402p
6
14580AK04.pgs 3/26/09 12:06 PM Page 325
29.
30.
31.
32. y
xO
1
21
22
23
2
p 2p
3
y
xO
1
21
2p3
p3
y
xO
2p 4p
1
21
y
xO
p
1
21
p2
326
33.
34.
35.
36.
37. y
xO
1
21p2
3p2
p22
y
xO
2p
1
214p
y
xO
p
1
21
p2
y
xO
2p
0.5
20.54p 6p
3p
y
xO
24
2p3
p3
4
14580AK04.pgs 3/26/09 12:06 PM Page 326
38.
Applying Skills
39. Since sine and cosine are cofunctions, it followsthat sin x 5 .
40. a.
b. 0.014 volt41. a–b.
c. The period of “middle C” appears to be one-half the period of C3.
42. a.
b. No. The period of this function is not 12months, but months. Extendingthis model would shift all the temperatures bymore than half a month each year.
2p0.5 < 12.566
f
t
726864
60
84
76
80
2 4 6 8 10
h
t
1
21
2 4 6 8
a.
b.
O10 12 14
t
0.014
20.014
0.05 0.1 0.15 0.2O
e
cos Ax 2 p2 B
y
xO
1
21
p4
9p4
5p4
327
Hands-On Activity1–2.
3. y 5 3 1 2 sin x4. Maximum 5 5, minimum 5 15. The amplitude is equal to one-half the difference
between the maximum and the minimum values:
6. (2) Graph(3) y 5 24 1 2 sin x(4) Maximum 5 –2, minimum 5 –6(5) The amplitude is equal to one-half the
difference between the maximum and
minimum values:
11-4 Writing the Equation of a Sine orCosine Graph (pages 457–459)
Writing About Mathematics1. No. The equation that Tyler wrote has period
p and phase shift p. Thus, it is equivalent to y 5 5 cos 2x. The maximum of this curve is at np and the minimum is at .
2. Yes. The phase shift of the first graph is equal tothe period of both equations.
Developing Skills3. a.
b.
4. a.
b.
5. a.
b.
6. a.
b.
7. a.
b.
8. a.
b. y 5 2 cos 3x
y 5 2 sin A3x 1 p2 B
y 5 cos 12(x 2 p)
y 5 sin x2
y 5 3 cos A2x 2 p2 B
y 5 3 sin 2x
y 5 2 cos x
y 5 2 sin Ax 1 p2 B
y 5 cos x
y 5 sin Ax 1 p2 B
y 5 cos Ax 2 p2 B
y 5 sin x
p2 1 np
A 5(22) 2 (26)
2 5 2
A 5 5 2 12 5 2
y
x
1
2122
4
23
Op2
3p2
5
14580AK04.pgs 3/26/09 12:06 PM Page 327
9. a.
b.
10. a.
b.
11. a.
b.
12. a.
b.
13. a.
b.
14. a.
b. y 5 2cos x
Applying Skills15. a. 0.75 m b. 10 s
c. 5 0.1 cycle per second
d.
e. No; if the amplitude is 0.75, then the maximumheight is 0.75 meter.
11-5 The Graph of the Tangent Function(pages 462–463)
Writing About Mathematics1. The tangent graph has no maximum or minimum
values, the period is p rather than 2p, it hasvertical asymptotes, and the range is all realnumbers rather than [21, 1].
2. No; the range is (2`, `).Developing Skills
3. Grapha. p b.c. (2`, `)
4. a–b.
c. 2
y
x
1
2122
4
23
Op2
3p2
2324
p
Ux: x 2p2 1 npV
h(t) 5 0.75 cos Ap5x B
110
y 5 2 sin Ax 2 p2 B
y 5 cos 2 Ax 1 p4 B
y 5 2 sin 2x
y 5 3 cos 12 Ax 2 p
4 By 5 3 sin
12 Ax 1 3p
4 By 5 2 cos 2x
y 5 sin 2 Ax 2 p4 B
y 5 2 cos Ax 2 p6 B
y 5 2 sin Ax 1 p3 B
y 5 cos Ax 2 p6 B
y 5 sin Ax 1 p3 B
328
5. a.
b–c.
d. They are the same.Applying Skills
6. a. h 5 r tan u b.7. a. (1)
(2)
(3) AP 5
(4) AB 5 s 5
(5) Perimeter 5
b. (1)
(2)
(3) AP 5
(4) AB 5 s 5
(5) Perimeter 5c. For any regular polygon with n sides
circumscribing a circle of radius r, theperimeter is .
11-6 Graphs of the Reciprocal Functions(pages 466–467)
Writing About Mathematics1. Cotangent is the reciprocal of tangent. As the
value of tan x increases, its reciprocal decreases,so cot x decreases for all values of x for which itis defined.
2nr tan pn
2(5)r tan p5
2r tan p5
r tan p5
p5
2p5
2(4)r tan p4
2r tan p4
r tan p4
p4
p2
V 5 13pr3
tan u
O
y
x1
p2
2 p2
2
21
22
O
y
x1
p2
2 p2
2
21
22
14580AK04.pgs 3/26/09 12:06 PM Page 328
2. sec x increases from 1 to ` in the interval
and increases from 2` to 21 in the
interval .
Developing Skills3. (3) 4. (8) 5. (6)6. (7) 7. (1) 8. (5)9. (2) 10. (4)
11. a.
b.
12. a.
b.
13. a.
b. 23p4 , 2p
4 , p4 , 3p4
O
y
x
1
p2
22p, 2p, 0, p, 2p
O
y
x21 p
2
23p2 , 2p
2 , p2 , 3p2
O
y
x
1
p2
Ap2 , pT
S0, p2 B
329
14.
15.
16.
17.18. y 5 cot x and y 5 csc x19. y 5 tan x and y 5 sec x20. a. Odd b. Odd c. Even d. OddApplying Skills21. a. a 5 10 sec u
b.
c. No d. 115.2 ft
11-7 Graphs of Inverse TrigonometricFunctions (pages 471–472)
Writing About Mathematics
1. Since sin (230°) 5 , . Thereference angle of 230° is 30°.
2. No, tan (220°) 5 0.839 � 1.Developing Skills
3. 30° 4. 60° 5. 45°6. 60° 7. 290° 8. 90°9. 260° 10. 135° 11. 245°
12. 245° 13. 0° 14. 180°15. 16. 0 17.
18. 19. 20.
21. 22. 23.
24. 0 25. 26. 027. 0 28. 0 29. 130. 0.5 31. 32. 20.5
In 33–35, part a, answers will be graphs.33. b.
34. b.
Range 5 5y: 21 # y # 16Domain 5 5x: 0 # x # p6y 5 cos xRange 5 5y: 0 # y # p6Domain 5 5x: 21 # x # 16y 5 arccos x
Range 5 5y: 21 # y # 16Domain 5 Ux: 2p
2 # x # p2 V
y 5 sin x
Range 5 Uy: 2p2 # y # p
2 VDomain 5 5x: 21 # x # 16y 5 arcsin x
2!22
p2
2p3
p3
2p3
p32p
3p3
p4
p2
arcsin A212 B 5 230821
2
22p, 2p, 0, p, 2p
23p2 , 2p
2 , p2 , 3p2
22p, 2p, 0, p, 2p
23p2 , 2p
2 , p2 , 3p2
u
a 10.2 10.6 11.5 13.1
2p9
p6
p9
p18
14580AK04.pgs 3/26/09 12:06 PM Page 329
35. b.
Applying Skills
36. a.
b. u 5 26.6° or u 5 0.46 radians or u 5 26° 349
c. 71° 349
11-8 Sketching Trigonometric Graphs(pages 474–475)
Writing About Mathematics
1. Yes. is the tangent graph with a
phase shift of . Therefore, the asymptotes alsohave a phase shift of .
2. No. . The
phase shift is , not .Developing Skills
3.
4. y
xO
p
3
23
p2
y
xO
2p
2
22
p
p4
p8
y 5 sin A 2x 2 p4 B 5 sin 2 Ax 2 p
8 Bp4
p4
y 5 tan Ax 2 p4 B
u 5 tan 21 A d
100 B
Range 5 5y: y is a real number6Domain 5 Ux: 2p
2 , x , p2 V
y 5 tan x
Range 5 Uy: 2p2 , y , p
2 VDomain 5 5x: x is a real number6y 5 arctan x
330
5.
6.
7.
8. y
xO
3
23
4p3
7p3
p3
y
xO
p
4
24
p2
y
xO
4p
2
22
2p
y
xO
1
212p3
p3
14580AK04.pgs 3/26/09 12:06 PM Page 330
9.
10.
11.
12. y
Ox
p
O
y
x
p2
2 p2
y
xO
4
24
3p
2pp
y
xO
1
21
p3
5p6
p62
331
13.
14.
15. a.
b.
16. a.
b.
17. a. y 5 4 sin x
b.
18. a.
b. y 5 4 cos x
19. a.
b.
20. a.
b.
21. a. b. 2y
xO
2
22
p2
y 5 2 cos Ax 1 p6 B
y 5 22 sin Ax 2 p3 B
y 5 4 cos Ax 1 p6 B
y 5 24 sin Ax 2 p3 B
y 5 4 sin Ax 1 p2 B
y 5 4 cos Ax 2 p2 B
y 5 22 cos Ax 2 p6 B
y 5 22 sin Ax 1 p3 B
y 5 cos Ax 2 p6 B
y 5 sin Ax 1 p3 B
y
xO
1
21p 2p
y
xO
2p
2
22
p
21
1
14580AK04.pgs 3/26/09 12:06 PM Page 331
22. a. b. 2
23. a. b. 4
Review Exercises (pages 476–479)1. a. 2 b. c.
d. {all real numbers} e. [22, 2]
f.
2. a. 3 b. 4p c.d. {all real numbers} e. [23, 3]
14p
y
xO
2
22
p3
2p3
21
1
3p2
2p3
y
xO
2
22
p2
y
xO
2
22
p2
3p2
332
f.
3. a. No amplitude b. p
c. d.e. {all real numbers}
f.
4. a. 1 b. p c.d. {all real numbers} e. [21, 1]
f.
5. a. 1 b. 2p c.
d. {all real numbers} e. [21, 1]
f. y
xO
1
212p p
12p
y
xO
1
21
p3
4p3
1p
y
x
p22 p
2
Ux: x 2p2 1 npV1
p
y
xO
1
21
22
23
2
2p
3
4p
14580AK04.pgs 3/26/09 12:06 PM Page 332
6. a. 2 b. 2p c.
d. {all real numbers} e. [22, 2]
f.
7. a. y 5 sin 2x
b.
8. a.
b.
9. a.
b.
10. a. y 5 2sin x
b.
11. (4) 12. (1) 13. (2)14. (3) 15. 16.
17. 18. 19.
20.
21. a. b. [21, 1]
c. d. Graph
22. a–b. c. 0
23. x 5 23p2 , x 5 2p
2 , x 5 p2 , x 5 3p
2
y
x
21
1
2p22p pp
S2p2 , p2 T
S2p2 , p2 T
2p6
p4
5p6
p6
p2
p6
y 5 cos Ax 1 p2 B
y 5 3 cos 12 Ax 2 p6 B
y 5 3 sin 12 Ax 1 5p6 B
y 5 2 cos Ax 1 p2 B
y 5 22 sin x
y 5 cos 2 Ax 2 p4 B
y
xO
2p
2
22
p
21
1
12p
333
24. a. b. 2
25. No. The function y 5 csc x is undefined at x 5 np
for integer values of n.
26. a.
b. 1 c. 25d. 110 mmHg e. 60 mmHg
27. a. 4.5 ft b. 14 hrc.
Exploration (pages 478–479)Answers will vary.
Cumulative Review (pages 479–481)Part I
1. 3 2. 4 3. 34. 4 5. 2 6. 17. 2 8. 1 9. 1
10. 3
Part II11.
�1 1 2 3 4 5 60
21 , x , 622 , 2x , 1227 , 2x 2 5 , 7
�2x 2 5� , 7
y 5 15.5 1 4.5 sin p7x
P
t
85
60
35
10
110
1 32
y
x
2
22
p21
1
2p
14580AK04.pgs 3/26/09 12:06 PM Page 333
12. Answer: 123°, 303°
Tangent is negative in the second and fourthquadrants.Therefore, and
.Part III13. Answer: x 5 0,
Therefore, x 5 0 is one solution.Use the quadratic formula to find the roots of thequadratic factor:
14.
5 94.5
1 3(2)2 1 3(2)3 1 3(2)4
a5
n50
3(2)n21 5 3(2)21 1 3(2)0 1 3(2)1
x 5 4 6 3ix 5
8 6 !2362
x(x2 2 8x 1 25) 5 0x3 2 8x2 1 25x 5 0
4 6 3i
u 5 360 2 57 5 3038
u 5 180 2 57 5 1238
u < 2578
u 5 tan21 (21.54)
tan u 5 21.54
334
Part IV
15. a. Since is the diagonal of a square,m�GBC 5 45.
b.
16. Answer: and (2, –3)
Substitute the linear equation into the quadraticand solve for x:
y 5 2(2) 2 7 5 23
y 5 2 A 12 B 2 7 5 26
x 5 12, 2
0 5 (2x 2 1)(x 2 2)0 5 2x2 2 5x 1 2
2x 2 7 5 2x2 2 3x 2 5y 5 2x 2 7
A 12, 26 B
6 u 5 tan 21 A !2
2 B < 358
tan u 5 GCAC 5 1
!2 5!22
BG
Chapter 12.Trigonometric Identities
12-1 Basic Identities (page 485)Writing About Mathematics
1. No. We also need to know the quadrant in whichthe terminal side of the angle lies to determinethe sign of the other trigonometric functions.
2. a. To derive 1 1 tan2 u 5 sec2 u, divide each termof sin2 u 1 cos2 u 5 1 by cos2 u. To derive cot2 u 1 1 5 csc2 u, divide each term of sin2 u 1 cos2 u 5 1 by sin2 u.
b. No; tan u and sec u are not defined forand cot u and csc u are undefined
for u 5 np, so the identities are not definedfor those values of u.
Developing Skills
3. 4. 5.
6. 7. 8.
9. 10. 11. sin u cos u
12. 13. 14. 1 1 sin u cos u cos u
2 cos u sin u
1 cos2 u
1 cos u
1 sin2 u
1 cos2 u
1 sin u
1 sin u
1 cos u
cos u sin u
sin u cos u
u 5 p2 1 np
12-2 Proving an Identity (pages 487–488)Writing About Mathematics
1. No. The equation is conditional. If u is an anglewhose terminal side lies in quadrant III or IV,then the equation is false.
2. Yes. The fraction is equal to 1. When the left sideis multiplied and simplified it becomes .
Developing Skills
3.
cos u 5 cos u ✔
4.
sin2 u 5 sin2 u ✔
5.
cos2 u 5 cos2 u ✔
cos usin u
1
? sin u1
1 ? cos u1 5? cos2 u
cot u sin u cos u 5? cos2 u
sin ucos u
1
? sin u1 ? cos u
1
1 5? sin2 u
tan u sin u cos u 5? sin2 u
sin u1
1 ? 1sin u
1
? cos u1 5? cos u
sin u csc u cos u 5? cos u
cos2 u1 1 sin u
14580AK04.pgs 3/26/09 12:06 PM Page 334
6.
✔
7.
✔
8. 9.
1 2 1 2
✔ ✔
10.
✔
11.
✔
12. 13.
✔ ✔
14. 15.
✔ ✔tan u 5 tan ucot u 5 cot u
sin ucos u 5? tan ucos u
sin u 5? cot u
5? tan u
1cos u
1sin u
5? cot u
1sin u
1cos u
sec ucsc u 5? tan ucsc u
sec u 5? cot u
cos u 5 cos usin u 5 sin u
5? cos u
cos usin u
1sin u
5? sin u
sin ucos u
1cos u
cot ucsc u 5? cos utan u
sec u 5? sin u
sin2 u 5 sin2 u1 2 cos2 u 5? sin2 u
cos u1
1 ? 1cos u
1
2 cos2 u 5? sin2 u
cos u sec u 2 cos2 u 5? sin2 u
cos u (sec u 2 cos u) 5? sin2 u
cos2 u 5 cos2 u1 2 sin2 u 5? cos2 u
sin u1
1 ? 1sin u
1
2 sin2 u 5? cos2 u
sin u csc u 2 sin2 u 5? cos2 u
sin u (csc u 2 sin u) 5? cos2 u
cos2 u 5 cos2 usin2 u 5 sin2 u
1 2 sin2 u 5? cos2 u1 2 cos2 u 5? sin2 u
5? cos2 usin u
1sin u
5? sin2 ucos u
1cos u
1 2 sin ucsc u 5? cos2 u1 2 cos u
sec u 5? sin2 u
1 1 sec u 5 1 1 sec u
1 1 1cos u 5? 1 1 sec u
1sin u
1
? sin u1
1 1 1sin u
1
? sin u1
cos u 5? 1 1 sec u
csc u sin u 1 csc u tan u 5? 1 1 sec u
csc u (sin u 1 tan u) 5? 1 1 sec u
1 2 csc u 5 1 2 csc u
1 2 1sin u 5? 1 2 csc u
1 cos u
1
? cos u1
1 2 1 cos u
1
? cos u1
sin u 5 ? 1 2 csc u
sec u cos u 2 sec u cot u 5? 1 2 csc u
sec u (cos u 2 cot u) 5? 1 2 csc u
335
16.
✔
17.
✔
18.
✔
19.
✔
20.
✔
21.
✔sec u 5 sec usec u 1 1 2 1 5? sec u
(sec u 1 1)(sec u 2 1)1
sec u 2 11
2 1 5? sec u
sec2 u 2 1sec u 2 1 2 1 5? sec u
tan2 usec u 2 1 2 1 5? sec u
sec u csc u 5 sec u csc u
sec u csc u 5? 1cos u sin u
sec u csc u 5? sin2 u 1 cos2 ucos u sin u
sec u csc u 5? sin2 ucos u sin u 1 cos2 u
sin u cos u
sec u csc u 5? sin ucos u 1 cos u
sin u
sec u csc u 5? tan u 1 cot u
1 2 sin u 5 1 2 sin u
cos2 u1
(1 2 sin u)cos2 u
1
5? 1 2 sin u
cos2 u (1 2 sin u)
1 2 sin2 u 5? 1 2 sin u
cos2 u1 1 sin u ? 1 2 sin u
1 2 sin u 5? 1 2 sin u
cos2 u1 1 sin u 5? 1 2 sin u
1 2 cos u 5 1 2 cos u
sin2 u1
(1 2 cos u)sin2 u
1
5? 1 2 cos u
sin2 u (1 2 cos u)
1 2 cos2 u 5? 1 2 cos u
sin2 u1 1 cos u ? 1 2 cos u
1 2 cos u 5? 1 2 cos u
sin2 u1 1 cos u 5? 1 2 cos u
cot u 5? cot u
cos usin u 5? cot u
cos2 usin u cos u 5? cot u
1 2 sin2 u sin u cos u 5? cot u
1sin u cos u 2 sin2 u
sin u cos u 5? cot u
1sin u cos u 2 sin u
cos u 5? cot u
tan u 5 tan u
sin ucos u 5? tan u
sin2 usin u cos u 5? tan u
1 2 cos2 usin u cos u 5? tan u
1sin u cos u 2 cos2 u
sin u cos u 5? tan u
1sin u cos u 2 cos u
sin u 5? tan u
14580AK04.pgs 3/26/09 12:06 PM Page 335
22.
✔
23.
✔
24. 25.
✔ ✔
26.
1
✔
27. It is undefined when sec u or csc u are undefined;that is, at where n is an integer.
12-3 Cosine (A 2 B) (pages 491–493)Writing About Mathematics
1. Yes, the equations were shown to be true for allreal numbers.
2. Yes. She used the identity cos (90 2 B) 5 sin Band she let �B 5 100°.
Developing Skills
3. 4. 5.
6. 7. 8. !3221
22!32
2!322
!2221
2
np2
1 5 1cos2 u 1 sin2 u 5? 1
5? 1sin u
1sin u
cos u1
cos u
cos usec u 1 sin u
csc u 5? 1
1 5 11 5 1
sin2 usin2 u
5? 1cos2 ucos2 u
5? 1
1 2 cos2 usin2 u
5? 11 2 sin2 ucos2 u
5? 1
1sin2 u
2 cos2 usin2 u
5? 11cos2 u
2 sin2 ucos2 u
5? 1
2 cos2 usin2 u
5? 11
sin usin u2 sin2 u
cos2 u 5? 1
1cos ucos u
csc usin u 2 cot2 u 5? 1sec u
cos u 2 tan2 u 5? 1
1 5 1
sin u 1 11 1 sin u 5? 1
sin u 1 sin2 u 1 cos2 u1 1 sin u 5? 1
sin u 1 sin2 u1 1 sin u 1 cos2 u
1 1 sin u 5? 1
sin u1 ? 1 1 sin u
1 1 sin u 1 cos2 u1 1 sin u 5? 1
sin u 1 cos2 u1 1 sin u 5? 1
1 5 1
cos u 1 11 1 cos u 5? 1
cos u 1 cos2 u 1 sin2 u1 1 cos u 5? 1
cos u 1 cos2 u1 1 cos u 1 sin2 u
1 1 cos u 5? 1
cos u1 ? 1 1 cos u
1 1 cos u 1 sin2 u1 1 cos u 5? 1
cos u 1 sin2 u1 1 cos u 5? 1
336
9. 10. 11.
12. 13. 14.
15. 16. 17. 21
Applying Skills
18. a. b.
c. d.
19. a. b.
c. d.
e. f.
20. a. b.
c. d.
e. f.
21. a. 45°b. 5 0.8
5 0.6
c.
d. 8°
12-4 Cosine (A 1 B) (pages 495–496)Hands-On Activity
3. Draw segment and let R be its point ofintersection with the x-axis.Then:
We have shown that cos u 5 cos (2u). Since RP 5 RP9, sin u 5 2sin (2u).
In steps 4–6, the procedures will be similar.Writing About Mathematics
1. No. Maggie added the angles, which is incorrect.The correct answer is:
2. Yes. See the answer to Exercise 1.
5 2 cos A cos B1 cos A cos (2B) 2 sin A sin (2B)
2 sin A sin B5 cos A cos Bcos (A 1 B) 1 cos (A 2 B)
cos (2u) 5 ORsin (2u) 5 2RPrcos u 5 ORsin u 5 RP
PPr
57!2
10
5 A 35 B Q!2
2 R 1 A 45 B Q!2
2 Rcos (u 2 458) 5 cos u cos 458 1 sin u sin 458
cos u 5 35
sin u 5 45
!6 1 !242
!6 1 !24
!6 1 !242
!6 1 !24
2122
!32
!6 2 !24
!6 2 !24
!2 2 !64
!6 2 !24
!3221
2
!6 1 !24
!6 1 !24
2!6 1 !2
4!6 1 !2
4
2!32
12
1221
22!22
2!32
!32
12
14580AK04.pgs 3/26/09 12:06 PM Page 336
Developing Skills
3. 4. 5.
6. 7. 8.
9. 10. 11.
12. 13. 14.
15. 16. 17.
Applying Skills
18. a. b. c.
19. a. b.
c. d.
20. a. b. c.
d.
21. a. AB 5 50, 5 0.6, 5 0.8
b.
c.
d. 280 ft
12-5 Sine (A 2 B) and Sine (A 1 B) (pages 498–500)
Writing About Mathematics1. No. William added the angles, which is incorrect.
The correct answer is:
2. Yes. See the answer to Exercise 1.Developing Skills
3. sin (180° 1 60°) 5
sin (180° 2 60°) 5
4. sin (180° 1 45°) 5
sin (180° 2 45°) 5 !22
2!22
!32
2!32
5 2 sin A cos B1 sin A cos (2B) 2 cos A sin (2B)
2 cos A sin B5 sin A cos Bsin (A 1 B) 1 sin (A 2 B)
AD 5 200!2 < 282.84 ft
!210 5 40
AD
cos (u 1 458) 5 ACAD
5!210
5 A 45 B Q!2
2 R 2 A 35 B Q!2
2 Rcos (u 1 458) 5 cos u cos 458 2 sin u sin 458
cos u 5 45sin u 5 3
5
5 cos 458
5 (1)(cos 458) 2 (0)(sin 458)5 cos 3608 cos 458 2 sin 3608 sin 458
cos 4058 5 cos (3608 1 458)
!6 1 !242
!22
!22
!2 1 !642
!2 1 !64
!3221
2
!6 2 !24
!2 2 !64
!6 2 !24
!6 2 !24
122
!32
!22
!3221
2
!22
!32
12
2!222
!3221
2
2122
!222
!32
337
5. sin (180° 1 30°) 5
sin (180° 2 30°) 5
6. sin (270° 1 60°) 5
sin (270° 2 60°) 5
7. sin (270° 1 30°) 5
sin (270° 2 30°) 5
8. sin (60° 1 90°) 5
sin (60° 2 90°) 5
9. sin (30° 1 90°) 5
sin (30° 2 90°) 5
10. sin (90° 1 60°) 5
sin (90° 2 60°) 5
11. sin (60° 1 270°) 5
sin (60° 2 270°) 5
12. sin (45° 1 270°) 5
sin (45° 2 270°) 5
13. sin (30° 1 270°) 5
sin (30° 2 270°) 5
14. sin (360° 1 60°) 5
sin (360° 2 60°) 5
15.
16.
17.
Applying Skills
18. a. b.
c. d.
19. a. b.
c. d.
e.
20. a. b.
c. d.
e. f. !6 1 !24
!2 2 !64
!2 2 !64
!6 2 !24
2!3221
2
2!6 1 !2
4
!6 1 !24
!6 1 !24
212
!32
!2 2 !64
!2 2 !64
!6 2 !24
!6 2 !24
sin Ap3 2 5p
4 B 5!2 2 !6
4
sin Ap3 1 5p
4 B 5 2!6 1 !2
4
sin A 2p3 2 p
6 B 5 1
sin A 2p3 1 p
6 B 5 12
sin A 3p2 2 2p B 5 21
sin A 3p2 1 2p B 5 21
2!32
!32
!32
2!32
!22
2!22
12
212
12
12
2!32
!32
212
12
2!32
2!32
212
212
12
212
14580AK04.pgs 3/26/09 12:06 PM Page 337
21. a.
b.
c. 12°
22. a.
b.
c.
23. Answer:
, ,
Therefore, A9 5 .
12-6 Tangent (A 2 B) and Tangent (A 1 B)(pages 502–504)
Writing About Mathematics1. If A or B is equal to 1 np for any integer n,
then tan A or tan B is undefined.2. When and , then tan A tan B 5 1.
That makes the denominator of the fraction zeroand the fraction undefined.
Developing Skills
3. tan (45° 1 30°) 5tan (45° 2 30°) 5
4. tan (45° 1 60°) 5tan (45° 2 60°) 5 22 1 !3
22 2 !3
2 2 !32 1 !3
B 5 p3A 5 p
6
p2
5 10Q7!210 R 5 7!2
5 10S A 45 B A !2
2 B 1 A 35 B A !2
2 B T5 10(sin a cos 458 1 cos a sin 458)10 sin (a 1 458)
5 10Q2!210 R 5 2!2
5 10S A 35 B A !2
2 B 2 A 45 B A !2
2 B T5 10(cos a cos 458 2 sin a sin 458)10 cos (a 1 458)
(10 cos (a 1 45), 10 sin (a 1 45))
sin a 5 45cos a 5 3
5r 5 "62 1 82 5 10
ArA2!2, 7!2Bx < 629.23 ft
x 513,000
5!3 1 12
5!3 1 1226 5 500
x
55!3 1 12
26
5 A 513 B Q!3
2 R 1 A 1213 B A 1
2 Bsin (u 1 308) 5 sin u cos 308 1 cos u sin 308
sin u 5 513, cos u 5 12
13
5 1499!2
5 A 1733 B Q2!2
3 R 2 Q20!233 R A 1
3 Bsin (x 2 y) 5 sin x cos y 2 cos x sin y
cos y 52!2
3
sin y 5 13
cos x 520!2
33
sin x 5 1733
338
5. tan (60° 1 60°) 5tan (60° 2 60°) 5 0
6. tan (180° 1 30°) 5
tan (180° 2 30°) 5
7. tan (180° 1 45°) 5 1tan (180° 2 45°) 5 21
8. tan (180° 1 60°) 5tan (180° 2 60°) 5
9. tan (120° 1 30°) 5tan (120° 2 30°) 5 undefined
10. tan (120° 1 45°) 5tan (120° 2 45°) 5
11. tan (120° 1 60°) 5 0tan (120° 2 60°) 5
12. tan (120° 1 120°) 5tan (120° 2 120°) 5 0
13. tan (240° 1 120°) 5 0tan (240° 2 120°) 5
14. tan (360° 1 60°) 5tan (360° 2 60°) 5
15.
16.
17.
Applying Skills
18.
19. 1 20.
21.
22.
tan (A 2 B) 5 243
tan (A 2 B) 52 2 (22)
1 1 (2)(22)
tan (A 2 B) 5 tan A 2 tan B1 1 tan A tan B
tan (A 1 B) 5 22.375
tan (A 1 B) 5 0.75 1 41 2 (0.75)(4)
tan (A 1 B) 5 tan A 1 tan B1 2 tan A tan B
tan A 5 sin A cos A 5 20.75
cos A 5 2"1 2 (0.6)2 5 20.8
274
5 tan u
5 0 1 tan u1 2 (0)( tan u)
tan (180 1 u) 5 tan 1808 1 tan u1 2 tan 1808 tan u
tan Ap3 2 p
4 B 5 2 2 !3
5 22 2 !3 tan Ap3 1 p
4 B tan A 5p
6 2 5p6 B 5 0
tan A 5p6 1 5p
6 B 5 2!3
tan Ap 2 p3 B 5 2!3
tan Ap 1 p3 B 5 !3
2!3!3
2!3
!3
!3
2 1 !3!3 2 2
2!33
2!3!3
2!33
!33
2!3
14580AK04.pgs 3/26/09 12:06 PM Page 338
23.
24. a. 1b. Yes, and .
25. a. b. 12°26. a. tan x 5 , tan y 5 b.
12-7 Functions of 2A (pages 507–508)Writing About Mathematics
1. No. Let 2u 5 A. Using the cofunction identity,cos A 5 sin (90° 2 A). Then by substitution,cos 2u 5 sin (90° 2 2u).
2. Yes. Let 2u 5 A. Using the quotient identity,. Then by substitution,
tan 2u 5 .
Developing Skills
3. a. b. c.
4. a. 1 b. 0 c. Undefined
5. a. b. c. 2
6. a. 1 b. 0 c. Undefined
7. a. b. c.
8. a. b. c.
9. a. b.
c. d. Quadrant I
10. a. b.
c. d. Quadrant IV
11. a. b.
c. d. Quadrant I
12. a. b.
c. d. Quadrant I
13. a. b.
c. d. Quadrant I
14. a. b.
c. d. Quadrant IV243
3524
5
12!1031
3149
12!1049
!3
12
!32
20!6
149
20!649
25!11
7
7182
5!1118
158
817
1517
!32122
!32
!312
!32
!3122
!32
!312
!32
sin 2ucos 2u
tan A 5 sin Acos A
1033
25
45
419
tan z 5 11 5 1tan (x 1 y) 5 1
tan (A 1 B) 5 0
223 1 2
3
1 2 A223 B A23 B
tan (A 1 B) 5
tan (A 1 B) 5 tan A 1 tan B1 2 tan A tan B
339
15. a. b.
c. d. Quadrant II
16. a. b.
c. d. Quadrant I
17. a. b.
c. d. Quadrant IV
18. a. b.
c. d. Quadrant III
19. a. b.
c. d. Quadrant III
20. a. b.
c. d. Quadrant II
21.
✔
22.
✔
23.
✔
24.
✔12 sec u csc u 5 1
2 sec u csc u
12 A 1
sin u B A 1cos u) 5? 12 sec u csc u
12 sin u cos u 5? 12 sec u csc u
1sin 2u 5? 12 sec u csc u
csc 2u 5? 12 sec u csc u
cos 2u 5 cos 2u
cos 2u 5 1 2 2 sin2 ucos 2u 5 (cos2 u 1 sin2 u) 2 (sin2 u 1 sin2 u)cos 2u 5 cos2 u 1 sin2 u 2 sin2 u 2 sin2 ucos 2u 5 cos2 u 2 sin2 u
csc u 5 csc u
1sin u 5? csc u
cos2 u 1 sin2 usin u 5? csc u
cos2 u 2 sin2 u 1 2 sin2 usin u 5? csc u
cos u1
(cos2 u 2 sin2 u) 1 sin u(2 sin u cos u1
)sin u cos u
1
5? csc u
cos u (cos 2u) 1 sin u (sin 2u)sin u cos u 5? csc u
cos 2usin u 1 sin 2u
cos u 5? csc u
cot u 5 cot u
cot u 5? cos usin u
cot u 5? 2 sin u cos u2 sin2 u
cot u 5? 2 sin u cos u1 2 (1 2 2 sin2 u)
cot u 5? sin 2u1 2 cos 2u
243
235
45
34
24523
5
125
2 513212
13
243
3524
5
2!145
59
2!149
2120119
2119169
120169
14580AK05.pgs 3/26/09 12:07 PM Page 339
25. a. Let 4A 5 2u, then 2A 5 u.
b. Let 4A 5 2u, then 2A 5 u.
c. Let 4A 5 2u, then 2A 5 u.
26. a.
b. m�BAC 5 2u 5
c.
27.
12-8 Functions of (pages 511–513)Writing About Mathematics
1. Yes. Cosine is positive in the first and fourthquadrants, that is, cos A . 0 when .
is in the first or fourth quadrant since
, so is positive.
2. Yes. Let . Using the quotient identity,
. Then by substitution,
.
Developing Skills
3. a. 2 b. 2 c.
4. a. b. c.
5. a. b. c.
6. a. 0 b. 21 c. 0
7. a. b. c. 21
8. a. b. c. 21
9. a. b. c.
10. a. b. c.
11. a. b. c. 2!5222
3!53
!1197
!4212
!10212
!77
!144
!24
2!22
!22
!222
!22
2!332
!32
12
!312
!32
!312
!32
sin 12A
cos 12Atan 12A 5
tan u 5 sin ucos u
12A 5 u
cos 12A2p4 , A
2 , p4
12A
2p2 , A , p
2
12 A
As A1, !15Bsin 2u 5 2Q!6
4 RQ!104 R 5
!154
cos 2u 5 10 2 616 5 1
4
cos u 5!10
4 , sin u 5!64 , r 5 4
x 5 r cos u, y 5 r sin u
1807 mi or 255
7 mi
718
56
tan 4A 5 2 tan 2A1 2 tan2 2A
tan 2u 5 2 tan u1 2 tan2 u
cos 4A 5 cos2 2A 2 sin2 2Acos 2u 5 cos2 u 2 sin2 u
sin 4A 5 2 sin 2A cos 2Asin 2u 5 2 sin u cos u
340
12. a. b. c.
13. a. b. c.
14. a. b. c.
15. a. b. c.
16. a. b. c.
17. a. 2 b. c. 2
18. a.
b.
c.
Applying Skills
19.
20.
21.
22.
23. a. .
b. .
É1 1 cos 12A
2cos 14A 5 6
cos 12u 5 6#1 1 cos u2
Let 14A 5 12u, then 12A 5 u
É1 2 cos 12A
2sin 14A 5 6
sin 12u 5 6#1 2 cos u2
Let 14A 5 12u, then 12A 5 u
5 $ 2 2 !3
4 5"2 2 !3
2
É1 2
!32
2sin 158 5
5 $(2 2 !3)2
1 5 2 2 !3
5 $ 2 2 !32 1 !3 ?
2 2 !32 2 !3
ä1 2
!32
1 1!32
tan 158 5
52 2 !2
!2 5 !2 2 1
5 $ (2 2 !2)2
2
5 $ 2 2 !22 1 !2
? 2 2 !22 2 !2
ä1 2
!22
1 1!22
tan 18 5
5 61 2 cos Asin A
5 6#(1 2 cos A)2
sin2 A
5 6#(1 2 cos A)2
1 2 cos2 A
5 6#1 2 cos A1 1 cos A 3 1 2 cos A
1 2 cos A
tan 12A 5 6#1 2 cos A1 1 cos A
21 1 !10
3
2"50 2 5!10
10
"50 1 5!1010
34
45
35
2122
2!55
!55
34
45
35
!242
2!2321
3
!622
!1052
!155
2!7323
4!74
14580AK05.pgs 3/26/09 12:07 PM Page 340
c. .
24. a.
b.
height of bill board 5 ft
Hands-On Activity: Graphical Support for theTrigonometric Identities
1. Yes2. Yes3. Yes4. Each graph, Y2 and Y3, coincides with Y1 only
part of the time. When is positive,Y2 coincides and when is negative,Y3 coincides. Neither Y2 nor Y3 is accurate for allvalues of x.
Review Exercises (page 514)1.
✔
2.
✔
3.
✔2 sin2 u 5 2 sin2 u
2 sin2 u 5? 1 2 (1 2 2 sin2 u)2 sin2 u 5? 1 2 cos 2u
csc u 5 csc u
1sin u 5? csc u
cos2 u 1 sin2 usin u 5? csc u
cos2 usin u 1 sin u A sin u
sin u) 5? csc u
cos u cos usin u 1 sin u 5? csc u
cos u cot u 1 sin u 5? csc u
sec u 5 sec u
sec u 5? 1cos u
sec u 5? 1sin u ? sin u
cos u
sec u 5? csc u tan u
cos x2
cos x2
60 2 503 5 431
3
h 5 503 or 162
3 ft
23 5 h
25
tan 12u 5height of base
25
5 #49 5 2
3
ä
1 2 513
1 1 513
tan 12u 5
cos u 5 2565 5 5
13
ä
1 2 cos 12A
1 1 cos 12Atan 14A 5 6
tan 12u 5 6#1 2 cos u1 1 cos u
Let 14A 5 12u, then 12A 5 u
341
4.
✔
5.
✔
6.
✔
7. 8. 9.
10. 11. 12.
13. 14. 15.
16. 17. 18.
19. 20. 21.
22. 23. 20.8 24.
25. 26. 27.
28. 29. 30.
31. 32. 33.
34. 35. 36. 7
37. 38. 39.
40. 41.
42. If A and B complementary,cos (A 1 B) 5 cos 90°.
✔sin A sin B 5 cos A cos B0 5 cos A cos B 2 sin A sin B
cos 908 5 cos A cos B 2 sin A sin Bcos (A 1 B) 5 cos A cos B 2 sin A sin B
2"50 1 35!2
102"50 2 35!2
10
724
2425
725
!210
7!210
5!272 9
16!77
!144
!243!7
18
3!78
!73
34
8!65
2!210
7!210
336527
725
3366252 44
117
43
117125
35
2 44125
45
34
72424
522425
tan A cot B 5 tan A cot B
sin Acos A ? cos B
sin B 5? tan A cot B
2 sin A cos B2 cos A sin B 5? tan A cot B
tan A cot B5?
sin A cos B 1 cos A sin B 1 sin A cosB 2 cos A sin Bsin A cos B 1 cos A sin B 2 sin A cos B 1 cos A sin B
sin (A 1 B) 1 sin (A 2 B)sin (A 1 B) 2 sin (A 2 B) 5? tan A cot B
tan u 5 tan u
sin ucos u 5? tan u
sin u(2 cos u 1 1)cos u(2 cos u 1 1) 5? tan u
2 sin u cos u 1 sin u2 cos2 u 2 1 1 cos u 1 1 5? tan u
sin 2u 1 sin ucos 2u 1 cos u 1 1 5? tan u
1 1 cos ucot u 5 1 1 cos u
cot u
tan u(1 1 cos u) 5? 1 1 cos ucot u
sin u(1 1 cos u)cos u 5? 1 1 cos u
cot u
sin ucos u 1 sin u cos u
cos u 5? 1 1 cos ucot u
sin ucos u 1 sin u 5? 1 1 cos u
cot u
tan u 1 1csc u 5? 1 1 cos u
cot u
14580AK05.pgs 3/26/09 12:07 PM Page 341
Exploration (page 515)1. The equations appear to be identities since when
each left side is graphed in Y1 and each right sideis graphed in Y2, the graphs of Y1 and Y2coincide.
2. sin (2A 1 A) 5 sin 2A cos A 1 cos 2A sin Asin (2A 1 A) 5 (2 sin A cos A) cos A
1 (2 cos2 A 2 1) sin Asin (2A 1 A) 5 2 sin A cos2 A 1 2 sin A cos2 A
2 sin Asin (2A 1 A) 5 sin A (4 cos2 A 2 1)sin (2A 1 A) 5 sin A (4 (1 2 sin2 A) 2 1)sin (2A 1 A) 5 sin A (3 2 4 sin2 A)
sin (3A) 5 3 sin A 2 4 sin3 A ✔
cos (2A 1 A) 5 cos 2A cos A 2 sin 2A sin Acos (2A 1 A) 5 (2 cos2 A 2 1) cos A
2 (2 sin A cos A) sin Acos (2A 1 A) 5 (2 cos2 A 2 1) cos A
2 (2 sin2 A) cos Acos (2A 1 A) 5 cos A [(2 cos2 A 2 1) 2 2 sin2 A]cos (2A 1 A) 5 cos A [2 cos2 A 2 1
2 2(1 2 cos2 A)]cos (2A 1 A) 5 cos A (4 cos2 A 2 3)
cos (3A) 5 4 cos3 A 2 3 cos A ✔
✔
Cumulative Review (pages 515–517)Part I
1. 4 2. 1 3. 24. 3 5. 4 6. 17. 1 8. 2 9. 3
10. 3
tan (3A) 5 3 tan A 2 tan3 A1 2 3 tan2 A
tan (2A 1 A) 52 tan A 1 tan A(1 2 tan2 A)
1 2 tan2 A 2 2 tan2 A
3 1 2 tan2 A1 2 tan2 A
A 2 tan A1 2 tan2 A
1 tan A BA 1 2 2 tan2 A
1 2 tan2 A Btan (2A 1 A) 5
2 tan A1 2 tan2 A
1 tan A
1 2 2 tan A1 2 tan2 A tan A
tan (2A 1 A) 5
tan (2A 1 A) 5 tan 2A 1 tan A1 2 tan 2A tan A
342
Part II
11.12.
If x2 1 5x 1 c 5 (x 1 a)2, then x2 1 5x 1 c 5 x2 1 2ax 1 a2.Therefore, 2a 5 5, so and .
Part III13. x2 1 3x 2 10 $ 10
(x 1 5)(x 2 2) $ 0x # 25 or x $ 2
14. center: (1, 2)
radius:
equation:
Part IV15. a.
b.
16. a.
b.
c. a8
n51
A!3Bn21 or a8
n51
3n 2 1
2
1, !3, 3, 3!3, 9, 9!3, 27, 27!3r 5 !39 5 r4
a5 5 a1r521
g(x) 5 3 1 sin Ax 1 p4 B
y
xO 1
p2
(x 2 1)2 1 (y 2 2)2 5 10
"(2 2 1)2 1 (5 2 2)2 5 !10
�3 �1 0 1 2 3 4�2�6 �4�5�7
a2 5 c 5 254a 5 5
2
a 5 52 and c 5 25
4
11 1 i 5 1
1 1 i ? 1 2 i1 2 i 5 1 2 i
2 5 12 2 1
2i
14580AK05.pgs 3/26/09 12:07 PM Page 342
Chapter 13.Trigonometric Equations
343
13-2 Using Factoring to SolveTrigonometric Equations (pages 529–530)
Writing About Mathematics1. No. The method of using factoring to solve a
trigonometric equation depends on themultiplicative property of zero: If ab 5 0, then a 5 0 or b 5 0. Thus, the right side of theequation must equal 0.
2. Yes. 2(sin u)(cos u) 1 sin u 1 2 cos u 1 1 5 0
sin u (2 cos u 1 1) 1 1(2 cos u 1 1) 5 0
(sin u 1 1)(2 cos u 1 1) 5 0
sin u 1 1 5 0 2 cos u 1 1 5 0Developing Skills
3. 30°, 150°, 270° 4. 30°, 150°, 210°, 330°5. 60°, 120°, 240°, 300° 6. 45°, 135°, 225°, 315°7. 60°, 300° 8. 90°, 210°, 270°, 330°9. 45°, 63.4°, 225°, 243.4° 10. 0°, 70.5°, 289.5°
11. 19.5°, 41.8°, 138.2°, 160.5°12. 66.4°, 113.6°, 246.4°, 293.6°13. 63.4°, 99.5°, 243.4°, 279.5°14. 70.5°, 75.5°, 284.5°, 289.5°15. 1.11, 1.25, 4.25, 4.39 16. 0, 1.82, 4.4617. 0, p (3.14), 3.55, 5.87 18. 3.48, 5.9419. 0.25, 0.52, 2.62, 2.89 20. 0.17, 1.11, 3.31, 4.2521. 22. 2.03
23. 290°, 270°
13-3 Using the Quadratic Formula toSolve Trigonometric Equations (page 534)
Writing About Mathematics1. When the discriminant is negative, the solutions
are imaginary numbers.2. When factored, and .The
range of the cosecant function is.
Developing Skills3. 202°, 338° 4. 74°, 125°, 254°, 305°5. 29°, 99°, 261°, 331° 6. 14°, 166°, 246°, 294°7. 111°, 159°, 291°, 339° 8. 46°, 80°, 280°, 314°9. 50°, 157°, 230°, 337° 10. 72°, 144°, 216°, 288°
11. 55°, 125° 12. 39°, 119°, 219°, 299°13. 64°, 140°, 220°, 296° 14. { }15. 16. 0.56, 5.72p
4 , 3p4 , 5p
4 , 7p4
(2`, 21g < f1, `)
csc u 5 12csc u 5 0
308 or p6
13-1 First Degree TrigonometricEquations (pages 524–526)
Writing About Mathematics1. The second equation simplifies to sin x 5 2 and 2
is outside the range of the sine function.2. The second equation simplifies to tan x 5 1.
Since the tangent function is periodic, there arean infinite number of x-values where tan x 5 1.
Developing Skills3. 60°, 300° 4. 150°, 330°5. 90° 6. 90°, 270°7. 135°, 315° 8. 45°, 315°9. 10. 0
11. 12.13. 14.15. 49° 16. 79°17. 71° 18. 24°19. 12° 20. 18°21. 75.5°, 284.5° 22. 16.6°, 163.4°23. 104.0°, 284.0° 24. 131.8°, 228.2°25. 0.17, 2.97 26. 2.09, 4.19 or ,27. 0.38, 3.52 28. 3.02, 6.16Applying Skills29. a.
b. 220 volts c. 2d. (1) u 5 0.93, 5.36
(2) t 5 0.30 s and 1.70 s30. a. x 5 36.9° b. u 5 18.4°31. a. We used the expression for the cotangent
function, which is undefined at .
b. Yes.
.sin Ap2 2 p
2 B 5 0
5 01 5 0 and
cos p2sin p2
cot Ap2) 5
p2
1tan u
E
t1
10
25
215
5
210
15
2O
4p3
2p3
p2 , 3p
2p6 , 5p
6
5p4 , 7p
4p4 , 5p
4
p3 , 2p
3
14580AK05.pgs 3/26/09 12:07 PM Page 343
13-4 Using Substitution to SolveTrigonometric Equations InvolvingMore Than One Function (pages 537–538)
Writing About Mathematics1. Yes. The maximum value of both sin u and cos u
is 1, and they are never equal to 1 for the samevalue of u. Therefore, their sum will always beless than 2.
2. 0° # u # 180°. Sine is positive in the first andsecond quadrants.
Developing Skills3. 30°, 150° 4. 30°, 150°5. 45°, 90°, 225°, 270° 6. 30°, 150°, 270°7. 0°, 60°, 300° 8. 30°, 150°, 210°, 330°9. 45°, 135°, 225°, 315° 10. 90°, 270°
11. 45°, 135°, 225°, 315° 12. 0°, 180°, 210°, 330°13. 30°, 150° 14. 45°, 135°, 225°, 315°Applying Skills15. a. (1) (2)
b. 0° c. 628.96°
13-5 Using Substitution to SolveTrigonometric Equations InvolvingDifferent Angle Measures (pages 540–541)
Writing About Mathematics1. No. Dividing by 2 divides the coefficients, not the
angles.2. No. You must account for the factor cos u. The
solution set also includes the numbers .Developing Skills
3. 30°, 90°, 150°, 270° 4. 0°, 180°, 360°5. 0°, 180°, 360°6. 30°, 90°, 150°, 210°, 270°, 330°7. 30°, 150°, 210°, 330° 8. 60°, 300°9. 30°, 90°, 150° 10. 45°, 135°, 225°, 315°
11. 12. 0, 1.82, p, 4.46, 2p
13. 14.
15. 0, 1.15, 1.99, p, 4.29, 5.13, 2p
16. 1.36, 4.92 17. 0.28, 2.86 18. 0.12, 3.02Applying Skills19. a.
b.c.
d. 49.92 mu 5 arctan 23 5 33.698
tan u 5 23
sin u 5 23 cos u
90 sin u 5 60 sin (908 2 u)d 5 60 sin (908 2 u)d 5 90 sin u
0.17, p2 , 2.97, 3p20.34, p2 , 2.80
p5 , 3p
5 , 7p5 , 9p
5
Up2 , 3p
2 V
2!2 2 2 in.2!3 2 2 in.
344
20. a.b.
, reject
Review Exercises (page 543)1. 120°, 240° 2. 240°, 300°3. 60°, 300° 4. 60°, 180°, 300°5. 0°, 180, 360° 6. 60°, 120°, 240°, 300°7. 45°, 135°, 225°, 315° 8. 45°, 135°, 225°, 315°9. 22.5°, 202.5°
10. 30°, 90°, 150°, 210°, 270°, 330°11. 3.43, 5.99 12. { }13. , 3.39, 6.03 14. 0, 2.30, 3.9815. 1.34, 2.91, 4.48, 6.05 16. 1.20, 1.43, 4.85, 5.0817. 0, 0.62, 2.53, p, 3.76, 5.6718. , 3.39, , 6.03 19. 1.33, 4.47
20. 1.23, , 5.05 21. 1.11, 1.77, 4.25, 4.91
22.
23. The left side of the equation is equal to zero onlyat values of u for which both the tangent andsecant functions are undefined.
However, tangent is undefined at .
24. a.
b.
c.
d.
e.
Exploration (page 544)(1) a. A 5 4 tan u b. 0.12(2) a. b. 0.24(3) a. b.
(4) a. b.
(5) a.
Total area 5 p2 2 1
2 sin u cos u
Area of semicircle 5 p2
212 sin u cos u5 12 sin (p 2 u) cos (p 2 u) 5
5 12bh
Area of triangle
p2A 5 1
2 sin u
p4A 5 1
2 sin 2u 5 sin u cos uA 5 2 tan u
m/A 5 248, m/B 5 488, m/C 5 1088
u 5 arctan !55 5 24.018
tan u 5 CDAD 5
!55
CD 5!55 AD
tan 2u 5 CDDB
tan u 5 CDAD
u 5 p2
u 5 p2
sin u 5 1tan u 2 sec u 5 sin u
cos u 2 1cos u 5 0
p3 , 5p
3
2p3 , 4p
3
3p2
p2
p2
u 5 arccos 78 5 28.968
cos u 5 1.752 5 7
8
sin u 5 0sin u (2 cos u 2 1.75) 5 02 sin u cos u 5 1.75 sin u
sin 2u 5 1.75 sin u sin u 5
yx, sin 2u 5
1.75yx 5
7y4x
14580AK05.pgs 3/26/09 12:07 PM Page 344
b. No possible value for u. Since the area of thesemicircle circle is square units, theminimum area of the shaded region is square units.
(6) a.b.
Cumulative Review (pages 545–546)Part I
1. 3 2. 1 3. 44. 1 5. 2 6. 37. 4 8. 4 9. 2
10. 1Part II11.
12.
f21 (x) 5 x 1 2
4
x 1 24 5 y
x 1 2 5 4yx 5 4y 2 2
y
xO
121
p12
A 5 2 sin u cos u 5 sin 2u
p2 . 1
p2 < 1.57
345
Part III
13. ,
14. Center 5
Radius 5
(x 2 1)2 1 (y 2 2)2 5 72Part IV
15.
16.
x 5 32 6 1
2i
x 56 6 !36 2 4(2)(5)
4
0 5 2x2 2 6x 1 5 5 0
22 5 5 2 6xx2
22 5 5x2 2 6
x
6x 2 2 5 5
x2
sec u 5 23!2
4
cot u 5 22!2tan u 5 2!24
cos u 5 22!2
3sin u 5 13
5 6!2
5 !36 1 36"(22 2 4)2 1 (5 1 1)2
A22 1 42 , 5 2 1
2 B 5 (1, 2)
2!3 81, 18, 18!3 9, 18!3 812, 2!3 9
Chapter 14.Trigonometric Applications
14-1 Similar Triangles (pages 551–552)Writing About Mathematics
1. Yes, since , u can be found byusing arctan.
2. Quadrant III. Both cosine and sine are negativewhen evaluated.
Developing Skills
3. 4.
5. (0, 6) 6.
7. 8. (20.5, 0)
9. 10.
11. (0, 212) 12. (21, 21)
13. 14. A1, 2!3BQ!32 , 23
2R
A2252 !3, 225
2 BA292!3, 92 B
A2152 !2, 15
2 !2 BA24, 4!3BA!3, 1BA2!2, 2!2B
tan u 5 28.4825.30 5 1.6
15. a. 10 b. 53°16. a. 13 b. 113°17. a. 7 b. 90°18. a. 15 b. 323°19. a. 15 b. 0°20. a. 14.42 b. 236°21. a. 25 b. 16°22. a. 11.66 b. 301°23. a. 11.31 b. 135°
24. a. R(5, 0), b. sq units
25. a. R(12, 0), S(0, 8) b. 48 sq units
26. a. R(8, 0), b. sq units
27. a. R(20, 0), S(10, 10) b. 100 sq units28. a. R(9, 0), b. sq units29. a. R(7, 0), b. 28 sq unitsSA8!3, 8B
20.25!3SA4.5, 4.5!3B
16!2SQ24!2, 4!2R
15!34SQ1.5, 3!3
2 R
14580AK05.pgs 3/26/09 12:07 PM Page 345
14-2 Law of Cosines (pages 555–556)Writing About Mathematics
1. Let C be the obtuse angle of �ABC. By the Lawof Cosines, c2 5 a2 1 b2 2 2ab cos C. Since thecosine of an obtuse angle is negative,2(2ab cos C) is positive. Therefore,c2 5 a2 1 b2 1 �2ab cos C�. A whole is greaterthan the sum of its parts, so c2 . a2 or c . a,and c2 . b2 or c . b.
2. The cosine of a right angle is zero. Thus,when c is the hypotenuse,c2 5 a2 1 b2 2 2ab cos C 5 a2 1 b2, which is the Pythagorean theorem.
Developing Skills3. m2 5 a2 1 r2 2 2ar cos M4. p2 5 n2 1 o2 2 2no cos P5. 6. 7.8. 4 9. 10.
11. 12. 13.14. 5.6 15. 147.0 16. 4.817. 98.6 18. 1.7 19. 7.5Applying Skills20. a. 0.72 mi b. 1.70 mi21. 24.08 lb22. a. 87 m b. 74 m23. 28 ft 24. 151.1 m25. 36.5 nautical miles26. °
14-3 Using the Law of Cosines to FindAngle Measure (pages 558–559)
Writing About Mathematics1. Let �C be the angle opposite the side of length
12. Then:
No �C exists such that cos �C is less than 21.2.
If is obtuse, is negative; thus.
Developing Skills
3. 4.
5. 20.575 6. 0
cos Q 5p2 1 r2 2 q2
2prcos T 5 u2 1 v2 2 t2
2uv
c2 . a2 1 b22ab cos C/C
c2 1 2ab cos C 5 a2 1 b2c2 5 a2 1 b2 2 2ab cos C
cos /C 5 27956 , 21
79 5 256 cos /C122 5 42 1 72 2 2(4)(7) cos /C
6 c 5 x5 x25 2x2 2 x2
c2 5 2x2 2 2x2 cos 60
2!109!72!19!13!268!2!372!7
346
7.
9.
11.
13. 33°, 64°, 83° 14. 36°, 40°, 104°15. 42°, 51°, 87° 16. 47°, 47°, 86°17. 48°, 63°, 69° 18. 16°, 74°, 90°19. 37° 20. 122°21. a. 33.7 in. b. 58°22. 83° 23. 82°24. Let x 5 the length of any side of the equilateral
triangle.Let �C 5 any angle of the triangle.
14-4 Area of a Triangle (pages 563–564)Writing About Mathematics
1. Since and are supplementary,sin A 5 sin (180° 2 A) 5 sin B.
2. Yes. The area of the rhombus is (PQ)(PS)(sin P).Since the sides are congruent,(PQ)(PS)(sin P) 5 (PQ)2(sin P).
Developing Skills3. 3 sq units 4. 30 sq units5. 60 sq units 6. 108 sq units7. 16.8 sq units 8. 12 sq units9. 77.5 sq units 10. 24,338.5 sq units
11. 12.6 sq units 12. 25,221.0 sq units13. 122.0 sq units 14. 36,615.3 sq units15. 16.17. 480 ft2
Applying Skills
18. a. b.
c. d. 6 km2
19. 234 ft2 20. 125!3 ft2
4!2 km2
2!23
13
36!2 cm2400!3 m2
/B/A
/C 5 608
cos /C 5 12
cos /C 5 x2
2x2
cos /C 5 x2 1 x2 2 x2
2x2
cos P 5 139160
cos N 5 95256
cos M 5 1180
cos F 5 6172
cos E 5 2948
cos D 5 2 17192
cos C 5 214
cos B 5 1116
cos A 5 78 8.
10.
12.
cos C 5 0
cos B 5 513
cos A 5 1213
cos R 5 2 516
cos Q 5 1320
cos P 5 3740
cos C 5 34
cos B 5 34
cos A 5 218
14580AK05.pgs 3/26/09 12:07 PM Page 346
21. sin 30° 5 sin (180° 2 30°) 5 sin 150°Thus, .
22. a. b. 41.8° or 138.2°
c. Yes, an acute or an obtuse triangle.23. a. A 5 ac sin u b. u 5 90°
14-5 Law of Sines (pages 567–568)Writing About Mathematics
1. No. A positive sine means the angle is in the firstor third quadrant; that is, the angle can be eitheracute or obtuse.
2. Yes. A positive cosine means the angle is in thefirst or fourth quadrant. However, in a triangle,each angle when drawn in standard position is inthe first or second quadrant. Thus, when thecosine is positive, the angle is acute. Similarly, anegative cosine means the angle is obtuse.
Developing Skills
3. 4. 48 5.6. 7. 12.5 8.9. 23.5 10. 31.4 11. 44.5
12. 18.3 13. 97.7 14. 16.915. 6.9316. a. 8.85 cm b. 32.2 cm17. a. 31.1 in. b. 83 in.18.
Applying Skills19. a. 3.18 ft b. 12.3 ft20. 138.0 ft, 250.2 ft21. a. 14.0 ft b. 18.6 ft22. 3.1 mi 23. $5,909
14-6 The Ambiguous Case (pages 573–574)Writing About Mathematics
1. Yes, the side opposite the largest angle, 110°, hasto be the largest side of the triangle. Since it isnot, no triangle can exist with thesemeasurements.
2. Because we know that one of the angles isobtuse, we cannot use the ambiguity of the sinefunction to imply two possible triangles.
Developing Skills3. a. 2 b. {20°, 155°, 5°}, {20°, 25°, 135°}4. a. 1 b. {30°, 60°, 90°}5. a. 1 b. {39°, 49°, 92°}6. a. 07. a. 1 b. {29°, 31°, 120°}
sin A 5 ac
asin A 5 c
1
asin A 5 c
sin 908
6434!64!33!6
23
12(AB)(BC) sin 308 5 1
2(DE)(EF) sin 1508
347
8. a. 1 b. {3°, 27°, 150°}9. a. 0
10. a. 2 b. {15°, 20°, 145°}, {5°, 15°, 160°}11. a. 012. a. 1 b. {135°, 30°, 15°}13. a. 1 b. {30°, 60°, 90°}14. a. 2 b. {45°, 62°, 73°}, {45°, 17°, 118°}Applying Skills15. a. 60.07°
b. No, the triangle formed by the ladder, wall,and ground is a right triangle.
16. Yes, there can be only one garden.Angles: {37°, 68°, 75°}Sides: {5 ft, 7.7 ft, 8 ft}
17. No. Since 10 , 12, 10 cm must be the length ofthe short diagonal. Therefore, the other anglemeasures 60°. Using the Law of Sines to find theangle opposite the 12 cm side yields a value ofsine greater than 1.
18. Yes. Two triangles are possible.Sides of 1st triangle: {2.0 km, 2.5 km, 2.7 km}Sides of 2nd triangle: {2.0 km, 2.5 km, 0.8 km}The route corresponding to the first triangle islonger.
14-7 Solving Triangles (pages 579–580)Writing About Mathematics
1. Since �BCD is a right triangle:
Use the Law of Sines in �BCA:
2. An angle of depression is the complement of thecomplement of the angle of elevation. Taking thecomplement of a complement is congruent to theoriginal angle.
Developing Skills3. a. Law of Cosines b. 4.94. a. Law of Sines b. 59.0°, 121.0°5. a. Law of Cosines b. 11.6°6. a. Law of Cosines b. 76.9°7. a. Both b. 115.7°8. a. Law of Cosines b. 122.6°9. a. Law of Sines b. 12.7°
10. a. Law of Sines b. 8.211.12.13.14.15. f 5 99, /D 5 438, /E 5 278
b 5 5, c 5 7, /A 5 908
a 5 34, c 5 31, /A 5 758
c 5 20, /A 5 258, /C 5 1258
c 5 17, /A 5 518, /B 5 698
BA < 98.62 ft
BAsin 408 5 148.193
sin 1058
BC 5 85sin 358 < 148.193 ft
14580AK05.pgs 3/26/09 12:07 PM Page 347
16.17.18.19.20.21.22.23. 27.4Applying Skills24. a. 78.6°, 101.4°
b. 37.8°, 40.8°, 60.6°c. 2.3 km
25. 107 ft 26. 16.6 ft, 19.2 ft27. 35 ft 28. 55.6°, 71.4°, 116.5°, 116.5°
Review Exercises (pages 582–584)1. 21 2. 17.5 in.3. 50° 4. 112°5.
6. a. 240 sq units b. 207. a. 31.3° b. 42.8 sq units8. AB = AC 5 26.2 9. 15, 21
10. a. 2b.c.
11.
12. a. 53°, 127° b. 25.0 in.c. Using the answers to part a: 626 in.2
13.
14.
15. 25.4 ft16. From A: 8.7 mi, from B: 7.0 mi17. a. b. 37°
c. Using the answer to part b: 13.9
Exploration (pages 584–585)Part AAnswers will vary. �DEF is an equilateral triangle.Part BSteps 1–8. Answers will vary.Step 9. Yes, �DEF is an equilateral triangle.
2425
m/B 5 1808 2 81.88 5 98.28
m/B 5 81.88
sin B 5 b sin Aa 5 8 sin 608
7 54!3
7
c 5 3, 50 5 c2 2 8c 1 15
49 5 64 1 c2 2 (16)(c) cos 608
sin R 5 r sin Pp 5 15 sin 668
12 5 1.14 . 1/B 5 1228, /C 5 108
/B 5 588, /C 5 748
6 u 5 90+
cos u 5 102 1 242 2 262
2(10)(24) 5 0
/A 5 258, /B 5 1358, /C 5 208
t 5 23, /R 5 408, /S 5 508
f 5 62, /E 5 908, /F 5 608
/P 5 378, /Q 5 538, /R 5 908
C 5 36, /A 5 288, /B 5 228
/R 5 1098, /S 5 448, /T 5 278
q 5 13, r 5 6, /P 5 708
348
Cumulative Review (pages 585–586)Part I
1. 2 2. 1 3. 44. 3 5. 1 6. 27. 3 8. 3 9. 3
10. 3Part II11.
12.
Part III13. Answer: x 5 2
x 5 6 x 5 2Reject extraneous root
14. Answer: {210°, 330°}
sin x 5
Reject extraneous rootPart IV
15.
16. a–b. c. 2y
xO
1
p2
x 5 12!2
x 53(16)2!2
logb x 5 logb A 3 ? 42
!8 B
x 5 2108, 3308
43sin x 5 21
2
Ax 1 12 B Ax 2 4
3 B 5 06 sin2 x 2 5 sin x 2 4 5 0
(x 2 6)(x 2 2) 5 0x2 2 8x 1 12 5 0x2 2 6x 1 9 5 2x 2 3
3 2 x 5 !2x 2 33 2 !2x 2 3 5 x
5 !5 2 32
5 6 2 2!524
5 1 2 2!5 1 51 2 5
1 2 !51 1 !5 ?
1 2 !51 2 !5
5 !6 1 !24
5 !22 ?
!32 1
!22 ? 1
2
5 cos 458 cos 308 1 sin 458 sin 308
cos 158 5 cos (458 2 308)
14580AK05.pgs 3/26/09 12:07 PM Page 348
15-1 Gathering Data (pages 594–595)Writing About Mathematics
1. A control group is necessary to ensure that anychanges to members of the experimental groupare due to the medication and not to someexternal factor, such as the placebo effect.
2. It is necessary that a participant does not knowto which group he or she belongs because thisknowledge can influence the participant’sperception of the effectiveness of the treatment.
Developing Skills3. Stem Leaf
9 0 0 2 58 2 4 5 5 6 7 87 4 5 6 86 6 7 7 85 4
4. Stem Leaf23 622 421 020 1 3 5 719 0 2 5 5 6 618 2 4 7 817 3 5 716 915 5
5. Stem Leaf15 114 0 1 2 2 3 4 4 713 0 1 1 2 2 4 6 7 7 7 7 8 912 7 9 9
6. No. ofBooks Read Frequency
8–9 16–7 34–5 52–3 90–1 7
Key: 12 � 9 � 129
Key: 15 � 5 � 155
Key: 5 � 4 � 54
349
7.
8.
9.
10.
11.
No. ofSiblings Frequency
6–7 14–5 22–3 110–1 16
Size Frequency15–17 312–14 11
9–11 76–8 5
Chapter 15. Statistics
1614
1210
86420
Fre
quen
cy
20
18
5–9 10–14 15–19 20–24 25–29 30–34 35–39
xi
14121086420
Fre
quen
cy
41–5051–60
61–7071–80
81–9091–100
101–110
xi
201612840
Fre
quen
cy
25–29 30–34 35–39 40–44 45–49 50–54 55–59
xi
14580AK05.pgs 3/26/09 12:07 PM Page 349
Applying SkillsIn 12–18, answers will vary.12. Record a sample of the temperature twice daily,
at perceived high and low temperatures, and takethe average over each month.
13. Conduct a survey on a sample, such as everytenth person leaving the restaurant on a givenday.
14. Conduct an observational study on a sample,such as recording the patient’s temperature every2 hours.
15. Conduct a census on the population, recordingall students’ grades on the test.
16. Conduct a census on the population, counting thenumber of people living in each house,apartment, etc. (Note that because of the size ofthe population, methods involving randomsamples will need to be used.)
17. Conduct a survey on a sample, such as measuringthe height of every fifth student enrolling inkindergarten in each elementary school.
18. Conduct a survey on a sample population ofmoviegoers, such as questioning every tenthperson leaving 100 randomly selected theaterswhere the movie is showing.
19. a. Stem Leaf9 2 3 5 8 88 2 3 4 4 6 6 7 7 8 97 2 4 5 6 7 76 1 6 5 3 8
b.
c. 21 d. 220. a. 2 b. 3 c. 12
d. 72 e. 10Hands-On ActivityAnswers will vary.
Score Frequency91–100 581–90 1071–80 661–70 251–60 2
Key: 5 � 3 � 53
350
15-2 Measures of Central Tendency (pages 604–605)
Writing About Mathematics1. No. Whenever the number of data values of a set
is odd, the number of data values less than thelower quartile or greater than the upper quartilecannot total exactly 50% of the number of datavalues.
2. Yes. Whether a set has 2n or 2n 1 1 data values,there are n data values above the median and ndata values below the median.
Developing Skills
3. Mean 5 , median 5 80, mode 5 804. Mean 5 66, median 5 65.5, mode 5 685. Mean 5 122.4, median 5 117, modes 5 115, 1186. Mean 5 2.4, median 5 2, modes 5 0, 27. Mean 5 $8.26, median 5 $7.88, mode 5 $7.508. Mean 5 $3.48, median 5 $3.50, mode 5 $5.009. Q1 5 6.5, median 5 15, Q3 5 21
10. Q1 5 36, median 5 42.5, Q3 5 4411. Q1 5 19, median 5 26, Q3 5 28.512. Q1 5 81, median 5 87, Q3 5 90.513. Q1 5 58, median 5 62, Q3 5 6614. Q1 5 19.5, median 5 26, Q3 5 3015. a. 90 b. 92 or 95
c. Any number other than 90, 92, or 9516. Q1 5 25.5, Q2 5 50.5, Q3 5 75.517. Q1 5 24.5, Q2 5 50, Q3 5 75.5Applying Skills18. a. 79.52 b. 82
c. Q1 5 74, Q3 5 89d.
19. a. 200.7 b. 202.5 c. 202d. Q1 5 191.5, Q3 5 208.5e.
20. 83Hands-On ActivityAnswers will vary.
81.1
191.5 202.5 208.5 223178
74 82 89 9947
14580AK05.pgs 3/26/09 12:07 PM Page 350
15-3 Measures of Central Tendency forGrouped Data (pages 611–614)
Writing About Mathematics1. Not necessarily. If the ages are not distributed
perfectly evenly, then Adelaide cannot make thisassumption.
2. Yes. There are only six possible ages thoseemployees could be, so there must be someemployees with the same age.
Developing Skills
3. Mean 5 , median 5 3, mode 5 34. Mean 5 32.4, median 5 30, mode 5 305. Mean 5 8.84, median 5 9, mode 5 96. Mean 5 , median 5 7, mode 5 77. Mean 5 $1.34, median 5 $1.30, mode 5 $1.308. Mean 5 , median 5 80, mode 5 859. 19th percentile 10. 16th percentile
11. 36th percentile 12. 32nd percentile13. Mean 5 12.8, median 5 12.614. Mean 5 78.5, median 5 78.515. Mean 5 $1.3, median 5 $1.316. Mean 5 11.3, median 5 10.917. Mean 5 $33.9, median 5 $27.5018. Mean 5 0.2, median 5 0.2Applying Skills19. Mean 5 11.625, median 5 12, mode 5 1220. Mean 5 17.43, median 5 1721. a. Mean 5 35.45, median 5 35
b. 25th percentile22. Mean 5 251.875, median 5 253.5823. Mean 5 48.8, median 5 51.125Hands-On ActivityAnswers will vary.
15-4 Measures of Dispersion (pages 617–619)
Writing About Mathematics1. No. The subscript for each data value indicates its
position in a list of data values, not its value.2. Yes. An outlier is a data value that is 1.5 times the
interquartile range below the first quartile orabove the third quartile. For the giveninformation, the interquartile range is 6, and 12 2 (1.5)(6) 5 3, which makes the data value 2an outlier.
Developing Skills3. Range 5 16
Interquartile range 5 104. Range 5 22
Interquartile range 5 8
81.6
6.63
3.08
351
5. Range 5 81Interquartile range 5 53
6. Range 5 40Interquartile range 5 4
7. Mean 5 35, median 5 35, range 5 30,interquartile range 5 10
8. Mean 5 7, median 5 7, range 5 8,interquartile range 5 2
9. Mean 5 24.9, median 5 25, range 5 26,interquartile range 5 17.5
Applying Skills10. a. Mean 5 2,296.5, median 5 480
b. The median is more representative. The meanis strongly influenced by the outlier.
c. The outlier is 19,014.d. With outlier removed, mean 5 439 and
median 5 427.e. The mean is more representative of the data
with the outlier removed.11. a. 75.6 b. 80
c. Q1 5 65, Q3 5 88.5d. 53 e. 23.5
12. Range 5 4, interquartile range 5 213. Range 5 9, interquartile range 5 2.514. a. 14 b. 3 c. 1515. Range 5 1, interquartile range 5 0.416. a. Range of A 5 9 5 range of B 5 9; yes, they
are the same.b. Post Office A:
Post Office B:
c. Interquartile range of A 5 1, interquartilerange of B 5 5
d. Post Office A. Wait times there of 9 and 10minutes are outliers, which is not the case atPost Office B.
Hands-On ActivityAnswers will vary.
2 3 9 101 5 64 87
2 3 9 101
* *
14580AK05.pgs 3/26/09 12:07 PM Page 351
15-5 Variance and Standard Deviation(pages 625–627)
Writing About Mathematics1. The second data set (from the sample) has the
larger standard deviation since its denominator issmaller.
2. Yes. If the standard deviation is the square rootof the variance, then the variance is the square ofthe standard deviation.
Developing Skills3. Variance � 3.92
Standard deviation � 1.984. Variance � 8.29
Standard deviation � 2.885. Variance � 116.74
Standard deviation � 10.806. Variance � 1,223.14
Standard deviation � 34.977. Variance � 66.24
Standard deviation � 8.148. Variance � 6.65
Standard deviation � 2.589. Variance � 233.36
Standard deviation � 15.2810. Variance � 32.99
Standard deviation � 5.7411. Variance � 877.38
Standard deviation � 29.6212. Variance � 12.57
Standard deviation � 3.5513. Variance � 648.99
Standard deviation � 25.4814. Variance � 106.78
Standard deviation � 10.3315. Variance � 3.23
Standard deviation � 1.8016. Variance � 4.20
Standard deviation � 2.05Applying Skills17. Line A. Since its standard deviation is smaller, its
late times are more closely clustered around themean of 10 minutes.
18. a. Variance � 269.43b. Standard deviation � 16.41
19. Variance � 0.66; standard deviation � 0.8120. 2.21 21. 2.14 22. 5.93Hands-On ActivityAnswers will vary.
352
15-6 Normal Distribution (pages 632–634)Writing About Mathematics
1. The mean of these scores is 90. In a normaldistribution, 50% of the scores are below themean. Only one of these five scores is below 90.
2. No. In a normal distribution the intervals closestto the mean contain more of the scores. Scoresare not uniformly distributed through the firststandard deviation above the mean.
Developing Skills3. 68% 4. 81.5% 5. 81.5%6. 84% 7. 84% 8. 50%9. 50%
10. a. 45 b. 52 c. 35 d. 28Applying Skills11. (4) 12. (2) 13. (3) 14. (3)15. a. About 0.62%. Use normalcdf(0, 16, 16.1, 0.04).
b. About 98.8%.Use normalcdf (16, 16.2, 16.1, 0.04).
16. About 1.3% of the time. Ken can expect to bepunctual approximately 98.7% of the time.This means he will be late approximately 100% 2 98.7% 5 1.3% of the time.
17. Approximately 91.04% of patrons check out fewer than 7 books.
18. 20.5 19. 820. The science test. On the math test, Nora’s score
was within 2 standard deviations of the mean. Onthe science test, her score was more than 3standard deviations above the mean.
15-7 Bivariate Statistics (pages 638–640)Writing About Mathematics
1. Univariate data consists of one number for eachdata point, or a single set of numbers. Bivariatedata consists of two numbers for each data point,or two different sets of numbers. Exampleanswers will vary.
2. A positive slope reflects a positive correlationand a negative slope reflects a negativecorrelation. Slope cannot be used to measure thestrength of a correlation.
Developing Skills3. Bivariate 4. Univariate5. Bivariate 6. Univariate7. Moderate linear correlation8. No linear correlation
14580AK05.pgs 3/26/09 12:07 PM Page 352
9. Strong linear correlation10. No linear correlationApplying Skills11. a.
b. Strong positive linear correlationc. y 5 30.714x 2 4.166
12. a.
b. Strong positive linear correlationc. y 5 2.365x 2 2.145
13. a.
b. Moderate positive linear correlationc. y 5 0.065x 2 7.681
353
14. a.
b. No linear correlation15. a.
b. Strong positive linear correlationc. y 5 0.765x 1 99.480
16. a.
b. Strong negative linear correlationc. y 5 20.965x 1 64.990
7654321
1 2 3 4 5 6
Family
Mot
her’
s fa
mily
00
330300270240210190160130
4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0
Gallons
Mile
s
5550454035302520
10 12 14 16 18 20 22 24
Number of ads
Sale
s
706560
26 28 30 32
161412108642
200 220 240 260 280 300 320 340
Calories
Fat
0360
240220
200180160140
120100
20 40 60 80 100
120
Seconds
Tem
pera
ture
300
280260
340320
360
140
160
180
200
220
240
260
280
300
320
3400
403530252015105
15 20 25 30 35 40 45 50
Age
% o
f spe
edin
g ac
cide
nts
0
5045
55 60 65
14580AK05.pgs 3/26/09 12:07 PM Page 353
17. a.
b. No linear correlation
15-8 Correlation Coefficient (pages 645–646)
Writing About Mathematics1. No. When �r� 5 1 there is a perfect linear
relationship between the data values, while acorrelation coefficient of 0 indicates no linearrelationship exists between the data values.
2. 1. There is a perfect linear relationship betweentemperature measured in degrees Fahrenheit andmeasured in degrees Celsius (otherwise theywouldn’t be measuring the same thing!).
Developing Skills3. 1 4. 05. 21 6. 07. Strong positive 8. Strong negative9. None 10. Moderate/weak positive
11. Strong positive 12. Moderate negative13. None 14. Strong negativeApplying Skills15. a.
450400350300250200150100
2 4 6 8 10 12 14 16
Gallons
Mile
s
550500
354
b. Close to 1. There appears to be a very strongpositive linear correlation.
c. r 5 0.9916. a.
b. Close to 21. There appears to be a very strongnegative linear correlation.
c. r 5 20.9617. a.
b. Close to 0. There does not appear to be astrong correlation.
c. r 5 0.3818. a.
b. Closer to 1. There appears to be a moderatepositive linear correlation.
c. r 5 0.75
7065605550454035
45 50 55 60 65 70
Same-day forecast
Act
ual t
empe
ratu
re
7654321
5 10 15 20 25 30 35 40
Total jobs
Hig
h pa
ying
jobs
0
1,700
1,600
1,500
1,400
1,300
1,200
140 150 160 170 180 190 200 210
Weight
Scor
e
1,800
1086420
2224
1 2 3 4 5 6 7 8
Month
Poun
ds lo
st
0 9 10 11
14580AK05.pgs 3/26/09 12:07 PM Page 354
19. a.
b. Closer to 1. There appears to be a moderatepositive linear correlation.
c. r 5 0.5920. a. 1. There would be a perfect positive linear
correlation.b. Greater than. Yes. Same-day forecasts should
be more accurate than forecasts for five daysin the future.
15-9 Non-Linear Regression (pages 651–654)
Writing About Mathematics1. The function y 5 ln x is undefined for x 5 0.2. Function y 5 axb has only positive or only
negative y-values when b is even. If b is odd, thepower function will have both positive andnegative y-values.
Developing Skills3. Quadratic 4. Exponential5. Logarithmic 6. Exponential7. Power 8. Cubic
9. a.
b. Quadratic. The scatter plot appears to bequadratic.
c. y 5 0.968x2 2 11.705x 1 40.950
161412108642
1 2 3 4 5 6 7 8
x
0 9 10
y
0
7570656055504540
45 50 55 60 65 70
Five-day forecast
Act
ual t
empe
ratu
re
0
908580
355
10. a.
b. Exponential. The scatter plot resembles anexponential curve. The curve does not passthrough the origin, yi . 0, and the y-interceptis positive.
c. y 5 8.609(1.560)x
11. a.
b. Cubic. The scatter plot resembles a cubiccurve.
c. y 5 0.291x3 2 0.027x2 1 0.557x 1 0.467
30
232629
212215218
22 21 0 1 2 3
x
23 4
y
221
96
24
12
1815
8070605040302010
22 21 0 1 2 3 4 5
x
23 6
y
0
10090
14580AK05.pgs 3/26/09 12:07 PM Page 355
12. a.
b. Logarithmic. The scatter plot resembles alogarithmic curve that does not pass throughthe origin, xi . 0, and the y-intercept appearsnegative.
c. y 5 26.995 1 2.003 ln x
13. a.
b. Exponential. The scatter plot resembles anexponential curve that does not pass throughthe origin, yi . 0, and the y-intercept ispositive.
c. y 5 3.127(0.859)x
Applying Skills14. a. y 5 999.843(1.045)x b. $1,623.0015. a. y 5 19.051 1 5.074 ln x b. 37.5 in.
356
16. a. The power regression equation,y 5 123,113.744x–1.981.
b. Yes. When Neptune’s orbital speed is pluggedinto the regression equation, we get 4,277.2million km as its distance from the sun, whichis a reasonably good estimate.
17. a.
b. Power regression. It resembles the positivehalf of a power function passing through (0, 0), xi . 0, and yi . 0.
c. y 5 2.024x2.991
18. a.
b. y 5 60.811x0.076
6766656463626160
1 2 3 4 5 6
Setting
Tem
pera
ture
(°F
)
0
706968
200175150125100755025
1 2 3 4 5 6
Height (ft)
Vol
ume
(ft3 )
0
275250225
0
2728
1 2 3 4 5 6 7 8x
0 9 10 11
y 2526
2324
22
0.7
1 2 3 4 5 6 7 8
x
0 9 10 11
y
0.91.11.31.51.7
2.3
1.92.1
2.52.72.9
14580AK05.pgs 3/26/09 12:07 PM Page 356
19.
The exponential regression model is y 5 0.407(1.294)x.No, Moore’s Law does not appear to hold forIntel chips. According to this model, the speed ofIntel computer chips increases by a factor of1.2942 � 1.67 every two years.
Hands-On Activity: Sine Regressiona.
b. y 5 23.442 sin (0.527x 2 1.713) 1 62.481c. 60°d. 72°
15-10 Interpolation and Extrapolation(pages 658–661)
Writing About Mathematics1. Interpolation is estimating a function value
between given values. Extrapolation is estimatinga function value outside the range of givenvalues.
2. A major source of error when usingextrapolation is that the regression model doesnot always hold outside of the given range ofvalues.
357
Developing SkillsIn 3–9, parts b and c, answers shown are obtainedusing the rounded regression equation from part a.
3. a. y 5 0.064x 1 0.971b. 1.336 c. 4.359
4. a. y 5 0.486x 1 2.607b. 8.439 c. 282.702
5. a. y 5 1.665x 2 1.147b. 20.315 c. 0.989
6. a. Logarithmic: y 5 0.800 1 0.413 ln xb. 0.939 c. 5.446
7. a. Quadratic: y 5 0.989x2 2 39.930x 1 407.290b. 70.546 c. 11.436, 28.938
8. a. Exponential: y 5 5.008(2.240)x
b. 23,837.928 c. 3.7139. a. Power: y 5 0.578x2.716
b. 493.156 c. 8.840Applying SkillsIn 10–16, answers shown are obtained using therounded regression equation.10. The linear regression model is
y 5 0.206x 2 400.971.a. 10.63% b. 14.12%
11. The linear regression model is y 5 0.041x 2 1.851.a. 5.53 sec b. 11.47 sec
12. The linear regression model is y 5 1.8x 1 32.a. 77°F b. 220°C
13. Let x 5 the number of years since 1980.The exponential model is y 5 78.753(1.187)x.a. 2,428 cars b. 56 cars
14. The power model is y 5 60.811x0.076.a. 65.2°F b. 4.35°F
15. The power model is y 5 2.000x3.000 or y 5 2x3.a. 3.91 ft3 b. 3.7 ft c. 7.4 ft
16. a. The logarithmic model is y 5 6.784 1 5.063 ln x.
b. 10.3 in. c. 22.0 in.
Review Exercises (pages 664–668)1. Univariate 2. Bivariate 3. Bivariate4. Census: counting data of general interest for an
entire population.Survey: asking questions (oral or written) to findout experiences, preferences, or opinions.Controlled experiment: structured study, usuallyof two groups, to compare results of a treatment or other process that only one groupundergoes.Observational study: structured study, usually oftwo groups, in which researchers do not imposethe treatment on either group.
7065605550454035
Month
Tem
p (°
F)
0
858075
1 2 3 4 5 6 7 8 9 10 11 12 13 14
3,5003,0002,5002,0001,5001,000
500
3 6 9 12 15 18 21 24
Year
Spee
d
00
27 30 33 36
14580AK05.pgs 3/26/09 12:07 PM Page 357
5. a. A sample. The student did not obtaininformation for every 9th grade student in thestate.
b. No. The data collected cannot be expected toreflect the grades of all students taking thetest. The sample was very small and was notrepresentative of the population as a whole,since the data was gathered from only onehigh school in the state.
6. a. 28 b. 80 c. 81d. Q1 5 79, Q3 5 86 e. 7f. Yes. The grade of 60 is an outlier since it is less
than 1.5 times the interquartile range belowthe lower quartile: 79 2 1.5(7) 5 68.5.
g.
7. a. 38.2 b. 38 c. 38d. 6 e. Q1 5 37, Q3 5 39f. 2 g. 2.43 h. 1.56
i. The menu on the calculator yields thesame values as those found in parts a–h.
8. The sample mean is 84.49 seconds.The sample variance is 1.957 seconds.a. 70% b. 96%c. The data appears to approximate a normal
distribution. The data appears bell-shaped,70% (close to the normal 68%) of the data iswithin one standard deviation of the mean,and 96% (close to the normal 95%) of thedata is within 2 standard deviations of themean.
9. a. Moderate negative linear correlationb. Negative
10. a. Strong positive linear correlationb. Positive
11. a. Strong negative linear correlationb. Negative
12. a. Moderate positive linear correlationb. Positive
STAT
358
13. a.
b. Yes, moderate positive linear correlation14. a. y 5 1.020x 1 0.024 b. r 5 0.99915. a. y 5 4x 1 47.5 b. r 5 0.97016. y 5 102.722(1.166)x
17. y 5 699.397 2 250.239 ln x 18. 24.4 million people 19. $67,50020. 351 dozen cookies21. a. 179 deer b. In the 7th year
Exploration1. y 5 13.619x2.122
2. y 5 35.938 1 1.627 ln x
Cumulative Review (pages 669–671)Part I
1. 2 2. 3 3. 24. 1 5. 4 6. 47. 1 8. 2 9. 3
10. 4Part II
11.
12. y
xO p 2p
2221
1
22p 2p
x 5 3 6 2ix 5 6 6 4i
2
x 52b 6 "b2 2 4ac
2a
x2 2 6x 1 13 5 0
70 78 80 8860 82 84 86
*
86420
2224
5 10 15 20 25 30 35 40
Stock price
Gai
n or
loss
0 45 50 55 6026
16141210
14580AK05.pgs 3/26/09 12:07 PM Page 358
Part III
13.
14.
Part IV15. Let T 5 the top of the tree.
Let C 5 the base of the tree.
x 5 33x 1 3 5 4x
33x13 5 34x(33)x11 5 (34)x
27x11 5 81x
5 13a 2 53b
5 13(a 1 b) 2 2b
5 13(log 2 1 log 3) 2 2 log 3
5 13(log 2 1 log 3) 2 log 32
log !3
69 5 1
3 log 6 2 log 9
359
To the nearest foot, the height of the tree is 57feet.
16.
Solutions: (0, 0) and (1, 1)
y
x
O1
1
TC < 56.71 ftsin 508 5 TC
AT
AT 5 74.033
20sin 108 5 AT
sin 408
m/ATB 5 180 2 130 2 40 5 108
m/BAT 5 180 2 50 5 1308
Chapter 16. Probability and the Binomial Theorem
16-1 The Counting Principle (pages 675–678)
Writing About Mathematics1. In the first situation, choosing a boy and choosing
a girl are independent events. In the secondsituation, the choice of the first girl affects thechoice of the second girl, and so the events aredependent.
2. The first is a dependent event, the second is anindependent event. In the first situation, thereare 52 3 51 5 2,652 possible outcomes. In thesecond, there are 52 3 52 5 2,704 possibleoutcomes.
Developing Skills3. 24 4. 120 5. 3366. 132 7. 256 8. 6259. 64 10. 125 11. 16 events
12. Independent 13. Dependent14. Dependent 15. Independent16. Independent 17. Dependent18. 216 19. 32 20. 6021. 48 22. 5,040 23. 3024. 465 25. 3,993,600 26. 12027. 12Applying Skills28. 6,720 29. 33630. 15,120 31. 2532. a. 462 b. 48433. 756
34. a. 72 b. 18 c. 3635. 25636. a. 24 b. 1237. a. 720 b. 12038. a. 4,096 b. 439. a. 720 b. 24040. a. 10,000 b. 5,040
c. Of the 10,000 telephone numbers with thisprefix, 5,000 form an even number.
41. 79 42. 69
16-2 Permutations and Combinations(pages 685–687)
Writing About Mathematics
1.
2.Developing Skills
3. 120 4. 479,001,600 5. 6,7206. 9 7. 604,800 8. 7209. 1,680 10. 720 11. 20
12. 4 13. 792 14. 79215. 210 16. 3,003 17. 3,00318. 1 19. 1 20. 12021. 1 22. 1 23. 72024. 180 25. 120 26. 3,36027. 40,320 28. 37,800 29. 50,40030. 4,989,600
n! 5 n(n 2 1)(n 2 2)c1 5 n(n 2 1)!
5 n!(n 2 r)! 3 r!
n!(n 2 r)!
r!nCr 5 nPrr! 5
14580AK05.pgs 3/26/09 12:07 PM Page 359
31. Order is not important and the chips are takenwithout replacement.a. 84 b. 1 c. 8,190d. 10,080 e. 1,170 f. 3,240g. 336 h. 756
32. 5,040 33. 15,120 34. 13,86035. 1,814,400 36. 4,200 37. 72038. 39. 10 40. 5
41. 10 42. 15 43. 12Applying Skills
44.
45. 56 46. 12047. 18,876 48. 14449. 50. 2,598,96051. 116,396,280 52. 1053. 3,02454. a. 5,040 b. c. 1
55. a. 927,048,304
b. 31,433,600
c. 92456. a. 720 b. 12057. 60,060
16-3 Probability (pages 691–694)Writing About Mathematics
1. a. Yes.
b. Empirical, since it is based on real data.2. Yes. Since the total probability of someone
getting the part must equal 1, the probability of itbeing Casey is 0.4.
Developing Skills
3. 4. 5.
6. a. 35,960 b. .000028
7. 8.
9. 10.
11. 12.
13. a. b. c. d.
14. a. b. c. d.
Applying Skills15. .416. Heads 5 526
1,000 5 .526, tails 5 4741,000 5 .474
720
1920
110
35
12
23
14
16
5 155
9!2!2!11!
2!2!2!
12180 5 1
15
212 5 1
61
5,525
1221
113
135,960 <
536
23
16
1,9542,000 5 .977 5 97.7%
96!(93!)3! ? 12!
(3!)(9!) 5
96!(90!)6! 5
17
10!4! 5 151,200
nCn2r 5 n!fn 2 (n 2 r)g!(n 2 r)! 5 n!
r!(n 2 r)! 5 nCr
12
360
17. a.
b. No. The theoretical probability for rolling a 5is . The die may be rigged.
18. 16%
19. a. Probability of exactly 2 plain:
b. Probability of exactly 1 maple, exactly 1 apple-
cinnamon:
c.
d.
20. 99.6%
21. a. � 0.3860
b. There is 1 way to choose Stephanie.There are 14 pairs involving Jan.There are possible choices.
Answer
22.
23. a.
b.
24. a.
b.
c. No, the empirical probability is much higherthan the theoretical probability. This is likelybecause many players have some skill andtherefore have a better than random chanceof hitting the bull’s-eye.
25. 16 in.2
26. a.
b.
c.
d. From the answer in part c, we can see that thecommon ratio is .
27. a.
b.
28. a. 5 0.25
b. 5 0.25416
416
20 C2 3 4C1
24C35
7602,024 5
95253 < .38
20 C3
24 C35
1,1402,024 5
285506 < .56
56
5n21
6n
56 ? 5
6 ? 56 ? 5
6 ? 16 5 54
65 < .0804
56 ? 5
6 ? 16 5 52
63 < .1157
721,270 < 0.0567
AbullAboard
5 p362 < 0.0024
18 C5
50 C55
15337,835 < 0.0040
48 C4
50C45
207245 < 0.8449
6C3
18 C35
1204,896 < 0.0245
1 3 14315 5 2
45 5 0.43C1 3 15C2 5 315
3C1 3 15C2
18C35
315816
10 C1 3 6C1 3 4C1
20C35
2401,140 < .2105
10 C2 3 6C1
20C35
2701,140 < .2368
6C1 3 4C1 3 10C1
20C35
2401,140 < .2105
10 C2 3 10 C1
20 C35
4501,140 < .3947
16 5 .16
4291,200 5 .3575
14580AK05.pgs 3/26/09 12:07 PM Page 360
16-4 Probability with Two Outcomes (pages 699–700)
Writing About Mathematics
1. . This is not a Bernoulli
experiment because each student is chosen
without replacement and so the choices are not
independent.2.
Developing Skills3. a. .15625 b. .3125 c. .3125
d. .15625 e. .03125 f. .03125g. Two or three heads
4. a. 0.4019 b. 0.1608 c. 0.0322d. 0.0032 e. 0.0001 f. 0.4019g. One or zero sixes
5. a. 0.2420 b. 0.0302 c. 0.0017d. 0.00004 e. 0.7260 f. No kings
6. a. 0.0584 b. 0.1877Applying Skills
7.
8. 3C1 (.2)1(.8)2 � .38409. 4C3 (.95)3(.05)1 � .1715
10. a. .2 b. 3C1 (.2)1(.8)2 5 .38411. 5C2 (.04)2(.96)3 5 .0142
12.
13. 5C5 (.92)5 (.08)2 5 .659114. 4C4 (.65)4 (.35)0 5 .1785
16-5 Binomial Probability and the NormalCurve (pages 706–708)
Writing About Mathematics1. No. Exactly r is included in both “at least” and
“at most,” so their sum will be greater than 1.2. No, as you can see from the histogram of the
probabilities, the graph is not bell-shaped.
0.25
0.20
0.15
0.10
0.05
0.00
Successes
Pro
babi
lity
0 2 3 4 5 6 7 8 9 10
0.40
0.35
0.30
1
0.45
20C2 A 112 B 2 A 11
12 B 185 .2755
7C5 A 23 B 5 A 1
3 B 2< .3073
No, nCr 5 n!r!(n 2 r)! 2
n!r! .
12 C2 3 8 C3
20 C55
3,69615,504 < .2384
361
Developing Skills
3. a. b. c. d.
4. a. b. c. d.
5. a. b. c. d.
6.
7.
8.
9.
10. .93 11. .49 12. .5413. .84; for the upper limit, use any value more than 3
standard deviations above the mean.14. .224 15. .45316. .176 17. .045
Applying Skills
18. 19. .9998 20. .4315
21. a. .0460 b. .105622. .382 23. .141 24. .655
16-6 The Binomial Theorem (pages 710–711)
Writing About Mathematics1. nCr 5 nCn–r
2. Yes,
.
Developing Skills3. x6 1 6x5y 1 15x4y2 1 20x3y3 1 15x2y4 1 6xy5 1 y6
4. x7 1 7x6y 1 21x5y2 1 35x4y3 1 35x3y4 1 21x2y5
1 7xy6 1 y7
5. 1 1 5y 1 10y2 1 10y3 1 5y4 1 y5
6. x5 1 10x4 1 40x3 1 80x2 1 80x 1 327. a4 1 12a3 1 54a2 1 108a 1 818. 16 1 32a 1 24a2 1 8a3 1 a4
9. 8b3 2 12b2 1 6b 2 110. 24 1 4i
an
i50nCi x
n22i5 an
i50nCi x
n2ix2i
Ax 1 1x B n
5 an
i50nCi x
n2i A 1x B i
12
a3
r5010Cr A 1
3 B r A 23 B 102r
a20
r5520Cr A 2
3 B r A 13 B 202r
a7
r5010Cr A 1
2 B r A 12 B 102r
5 a7
r5010Cr A 1
2 B 10
a15
r51015Cr A 1
2 B r A 12 B 152r
5 a15
r51015Cr A 1
2 B 15
609625
513625
297625
35
91216
227
215216
2527
3132
1316
12
316
14580AK05.pgs 3/26/09 12:07 PM Page 361
11. 11 1
1 2 11 3 3 1
1 4 6 4 11 5 10 10 5 1
1 6 15 20 15 6 11 7 21 35 35 21 7 1
1 8 29 56 70 56 29 8 11 9 37 85 126 126 85 37 9 1
12. 15C2x13y2 5 105x13y2
13. 10C6x4 y6 5 210x4y6
14. 6C3(2x)3(y)3 5 160x3y3
15. 9C4(x)5(2y)4 5 126x5y4
16. 7C5(3a)2(2b)5 5 6,048a2b5
17.
18. 6th term, 7th term 5 1,792a2b6
19. 4th term, 5th term 5 16d4
20. 5th term, 6th term 5 2231x6y5
Applying Skills21. 27x3 2 27x2 1 9x 2 1
22.
23.
Review Exercises (pages 713–714)1. 720 2. 120 3. 14. 20,160 5. 28 6. 17. 1 8. 210 9. 50
10. 132 11. 792 12. 4013. 40,320 14. 19,958,400 15. 36016. 120 17. 5618. a. .2036 b. .0011
c. .7738 d. .188719. a. b.
c.
20. .86821. a. .021 b. .023 c. .68322. x4 1 4x3y 1 6x2y2 1 4xy3 1 y4
23. 128a7 1 448a6 1 672a5 1 560a4 1 280a3 1 84a2
1 14a 1 124. 729 2 1,458x 1 1,215x2 2 540x3 1 135x4 2 18x5
1 x6
25.
1 28b4 2 8
b6 1 1b8
b8 2 8b6 1 28b4 2 56b2 1 70 2 56b2
4,0834,096 < .9968
794,096 < .019333
2,048 < .0161
75,000a5
i505Ci(1)52i(2.20)i 5 75,000a
5
i505Ci(2.20)i
100a12
i5012Ci(1)122i(.01)i 5 100a
12
i5012Ci(.01)i
8C3(y)5 A21y B 3
5 256y2
362
26. 24 2 4i27. 9C3 x
6(2y)3 5 284x6y3
28. 10C6 a4(3)6 5 153,090a4
29. 12C6 (2x)6(21)6 5 59,136x6
Exploration (pages 714–715)a. This calculation assumes that 7 games will be
played and is the probability that the AmericanLeague team will win 4 games. The World Seriesis won by the first team to win 4 games, and so asfew as 4 games may be played.
b. 4C4 p4 5 p4
c. 4C1 p4(1 2 p) 5 4p4(1 2 p)
d. 5C2 p4(1 2 p)2 5 10p4(1 2 p)2
e. 6C3 p4(1 2 p)3 5 20p4(1 2 p)3
f. p4 1 4p4(1 2 p) 1 10p4(1 2 p)2 1 20p4(1 2 p)3
Cumulative Review (pages 715–717)Part I
1. 4 2. 3 3. 44. 1 5. 4 6. 17. 4 8. 1 9. 3
10. 2
Part II
11.
12. Answer: x 5 1
Check x 5 1 Check x 5
✔ ✘12 2 21
22 5 2
#14 5? 21
2!4 5? 2
#3 A214 B 1 1 5? 2 A21
4 B!3(1) 1 1 5? 2(1)
214
x 5 1, 214
0 5 (x 2 1)(4x 1 1)0 5 4x2 2 3x 2 1
3x 1 1 5 4x2!3x 1 1 5 2x
x 5 822 5 164
22 5 log8 xlog2
14 5 log8 x
14580AK05.pgs 3/26/09 12:07 PM Page 362
Part III13. a–b.
14. Answer:
The solutions to the corresponding equality areand 4.
The original inequality is true in the interval.25
2 , w , 4
252
(2w 1 5)(w 2 4) , 0
2w2 2 3w 2 20 , 0
A 5 lw 5 2w2 2 3w , 20
l 5 2w 2 3
252 , w , 4
f21(x) 5 ln xln (1.6) 5 log1.6 x
y
xO 1
1
363
Part IV15. a.
b. c. 103
16.
u 5 2018, 3398
u 5 arcsin Q1 2 !32 R
sin u 5 12 6
!32
sin u 522 6 !4 1 8
24
22 sin2 u 1 2 sin u 1 1 5 0
2(1 2 sin2 u) 1 2 sin u 2 1 5 0
2 cos2 u 1 2 sin u 2 1 5 0
y 5 31.1327 1 31.1523 ln x
12010080604020
2 4 6 8 10
Week
Vol
unte
ers
00
14580AK05.pgs 3/26/09 12:07 PM Page 363
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