Un+7 B S cm Cam lOm II, Practise 1. c For help with questions I and 2, see Exam pies For each composite figure, • solve for any unknown lengths • determine the perimeter I and 2. Round to the nearest unit, if necessary. a) 13m b) Gm c) Sm A Scm 1 I lam 5 cm d) Scm 5cm *‘ ----z e) 12m 15cm cm 2. 6 cm a) Calculate the area of each composite figure. Round to the nearest square unit, if necessary. b) c) 30mm 6cm 4cm d) cm S cm e) 1) t 10cm ao cm 432 MHR’ChapterS
Transcript
Un+7 B
S cm
Cam
lOm
II,
Practise
1.
cFor help with questions I and 2, see Exampies
For each composite figure,• solve for any unknown lengths• determine the perimeter
I and 2.
Round to the nearest unit, if necessary.
a) 13m b)
Gm
c)SmA
Scm
1 I
lam
5 cm
d) Scm 5cm
*‘ ----ze)
12m
15cm
cm
2.
6 cm
a)
Calculate the area of each composite figure.Round to the nearest square unit, if necessary.
b)
c)
30mm
6cm
4cm
d)cm
S cm
e) 1)
t
10cm
ao cm
432 MHR’ChapterS
connect and Apply
3. a) What length of fencing is needed to surround this yard, to thenearest metre?
b) What is the area of the yard?
c) Explain the steps you took to solve this problem.
4. patrick is planning a garage sale. I-Is is painting six arrow signs todirect people to his sale.
a) Calculate the area of one side of one arrow.
b) Each can of paint can cover 2 m2. How many cans of paint willPatrick need to paint all six signs?
c) If the paint costs $3.95 per can, plus 8% PST and 7% CST, howmuch will it cost Patrick to paint the six signs?
5. Arif has designed a logo of her initial as shown. Use a ruler to makethe appropriate measurements and calculate the area of the initial, tothe nearest hundred square millimetres.
6. Create your own initial logo similar to the one in question 5.
Calculate the total area of your logo.
7. Use Technology
a) Use The Geometer Sketchpad® to draw your design fromquestion 0.
b) Use the measurement feature of The Geometer’s Sketchpad®to measure the area of your design.
8. Chapter Problem One of the gardens Emily is designing is madeup of two congruent parallelograms.
a) A plant is to be placed every 20 cm around the perimeterof the garden. Determine the number of plants Emily needs.
b) Calculate the area of her garden.
9. Use Technology Use The Geometer’s Sketchpad® to create acomposite figure made up of at least three different shapes.
a) Estimate the perimeter and area of the figure you created.
b) Determine the area using the measurement feature ofThe Geometer’s Sketchpad®. Was your estimate reasonable?
urn
iBm
16m
10 cm
60cm;75 cm
20cm
Sm
8.2 Perimeter and Area of Composite Figures MHR 433
a) Find the area of the outer ring, tothe nearest square centimetre.
b) What percent of the total area isthe outer ring?
11. The area of a square patio is 5 m2.a) Find the length of one of its sides, to the nearest tenth of a metre.b) Find the perimeter of the patio, to the nearest metre.
12. Brandon works as a carpenter. He is framing arectangular window that measures 1.5 m by 1 m.The frame is 10cm wide and is made up of fourtrapezoids. Find the total area of the frame, to thenearest square centimetre.
! Achievement Check
• 13. Susan is replacing the shingles on her roof. The roof is made up of ahorizontal rectangle on top and steeply sloping trapezoids on eachside. Each trapezoid has a (slant) height of 4.5 m. The dimensions ofthe roof are shown in the top view.
a) Calculate the area of the roof.b) A package of shingles covers 10 in2. How many packages
will Susan need to shingle the entire roof?c) Describe an appropriate way to round the number of packages
in part b).
1!¶1
10. An archery target has a diameter of80 cm. It contains a circle in the centrewith a radius of 8 cm and four additionalconcentric rings each 8 cm wide.
,
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434 MHR Chapter 8
xten d
14.
15.
Sanjav is designing a square lawn to fit inside a square yard wi (Iiside length 10 m so that there is a triangular flower bed at eachcorner.
a) Find the area of Sanjay’s lawn.
b) 1-low does the area of the lawn compare to the area of the flowerbeds?
c) Sanjay’s design is an example of a square inscithod withina square. The vertices of the inside square touch the sides ofthe outside square but do not intersect. WiJIl your answer inpart b) always be true when a square is inscribed vi thin a square?Explain
How does doubling (he radius of a circle affect its area? Justify your
a) Del ermine (he pattern rule for this sequence. and list the nextfour terms.
b) Construct rectangles using consecutive terms for (lie sides.The first rectangle is 1 by 1, the second is 1 by 2, the third is2 by 3, and so on. Find the area of each rectangle.
c) Explore the ratios of the sides of the rectangles. Make conject tires
about this ratio.
d) Explore the ratios of the areas of the rectangles. Make conjecturesabout this ratio.
17. Math Contest Determine the ratio of the perimeter of the smallestsquare to the perimeter of the largest square.
30cm
18. Math Contest The midpoints of the sides of a rectangle that measures10 cm by 8 cm are joined. Determine the area of the shaded region.
4 >lom
NiOn
answer using algebra.
16. Leonardo of Pisa lived in the 13th century in l’isa. Italy. He wasgiven the nickname Fibonacci because his fathers name x’asBonacci. Among his mathematical explorations is the sequence ofnumbers 1, 1, 2, 3, 5, 8, 13. 21
4cm
Scm
10cm
fri8.2 Perimeter and Area of Composite Figures’ MHR 435
practise
For help with question 1, see Example 1.
For help with question 2, see Exampie 2
2. Determine the volume of each object. Round to the nearest
cubic unit, when necessary.
7N.15 mmj /L - -\ :
I’20mm
V.
5 cm
4. Determine the volume of each object.
a) m/1
1OCJ j/6cm
8 cm
b)
5. A rectangular prism has length 3 m, width 2 m, and height 4 in.
a) Determine the surface area of the prism
b) Determine the volume of the prism.
Connect and Apply
6. A cereal box has a volume of 3000 cm3. If its length is 20 cm
and its width is 5 cm, what is its height?
a)1. Determine Lhe surface area of each object.
b)
12.m4
8.5 cm8.5
/
a) b) / \
2.6 mj// \ \\
For help ii’ith questions 3 to 5, see Example 3
3. Determine the surface area of each object
a) b)
8 mm
10mm 8 cm
8.3 Surface Area and Volume of Prisms and Pyramids’ MHR 441
ia. Chapter Problem The diagram shows the side view of the swimmingpool in Emily’s customer’s yard. The pooi is 4 m wide.
12m1m________
7. Sneferu’s North Pyramid at Dahshur, Egypt, is shown.Its square base has side length 220 m and its height is 105 m.
p
a) Determine the volume of this famous pyramid.
b) Determine its surface area, to the nearest square metre.
8.The Pyramid of Rhafre at Giza, Egypt, was built bythe Pharaoh Khafre, who ruled Egypt for 26 years.The square base of this pyramid has side length215 m and its volume is 2 211 096 m3.Calculate its height, to the nearest tenthof a metre
9. The milk pitcher shown is a right prism. The base hasan area of 40 cm2 and the height of the pitcher is 26 cm.Will the pitcher hold 1 L of milk?
10. A juice container is a right prism with a base area of 100 cm2.
a) If the container can hold 3 L of juice, what is its height?
b) Describe any assumptions you have made.
11. Adam has built a garden shed in the shapeshown.
a) Calculate the volume of the shed, to thenearest cubic metre
b) Adam plans to paint the outside of the shed,including the roof but not the floor. One canof paint covers 4 m2. How many cans
of paint wilt Adam need?
2m
I
m4m
c) If one can of paint costs S16.95, what is the total7% CST and 8% PST?
cost, including
t3m
4—3m—
a) Estimate how many litres of water the pooi can hold.
b) Calculate how many litres of water the pool can hold.
I
Ic) When the pool construction is complete, Emily orders water to.
fill it up. The water tanker can fill the pool at a rate of 100
How long will it take to fill the pool at this rate?
442 MHR’Chapter8
13. A triangular prism has a base that is a right triangle with shorter
sides that measure 6 cm and 8 cm. The height of the prism is 10 cm.
a) Predict how doubling the height affects the volume of the prism.
b) Check your prediction by calculating the volume of the original
prism and the volume of the new prism.
Was your prediction accurate?
d) Is tins true in general? If so, summarize the result.
Seasoong ant Proving
—Rep50000ire SeocoLr Tools
Proolern Solvire
CornecolngPrtlecOrnc
CommuncatLr
15. A pyramid and a prism have congruent square bases. If their
volumes are the same, how do their heights compare? Explain.
16. A statue is to be placed on a frustum of a pyramid. The frustum
is the part remaining after the top portion has been removed by
making a cut parallel to the base of the pyramid.
a) Determine the surface area of the frustum.
b) Calculate the cost of painting the frustum with gilt paint that
costs $49.50/rn2. It is not necessary to paint the bottom of the
frustum.
17. A formula for the surface area of a rectangular prism is
£4 = 2(]w ± vh ± Hi).
a) Suppose each of the dimensions is doubled. Show algebraically
how the surface area is affected.
b) How is the volume affected if each of the dimensions is doubled?
Justify your answer algebraically.
1 8. Math Contest A large wooden cube is made by glueing together
216 small cubes. Cuts are made right through the large cube along
the diagonals of three perpendicular faces.
How many of the small cubes remain uncut?
Achievement Check
14. a) Design two different containers that hold 8000 cm3 of rice. One
should be a rectangular prism and one should be a cylinder.
b) Determine the surface area of each one, to the nearest square
centimetre.
and wily?c) Which shape would you recommend to the manufacturer
Extend
3m
4m4m
8.3 surface Area and Volume of Prisms and PyramWs• MHR 443
communicate Your Understanding
a, A cone is formed from a circle with a 90°
sector removed. Another cone is formed
from a semicircle with the same radius.
How do the two cones differ? How are
they the same?
A cone is formed from a circle of
radius 10 cm with a 60° sectorremoved. Another cone is formed
from a circle of radius 15 cm with a
60° sector removed. How do the two
cones differ? How are they the same?
The slant height of a cone is doubled. Does this double
area of the cone? Explain your reasoning.
• PractiseFor help with questions I and 2, see the Example.
the surface
1. Calculate the surface area of each cone. Round to the nearest
square unit.
2. a)
b)
Find the slant height of the cone,
Calculate the surface area of the
cone. Round to the nearestsquare metre.
Connect and Apply
3. Some paper cups are shaped like cones.
a) I-low much paper, to the nearest square
centimetre, is needed to make the cup?
b) What assumptions have you made?
a) b)
Amc)
/\o cm
10cm
8.4 cm
-- 3.7 cm
///1am\
—Scmi-.
12cm
8.4 Surface Area of a Cone• MHR 447
R,a,onIng and Provinc
RL’prnsenrn S&ecilng TooIn
( PrbIL’t71
.flectthg
ComnunIcatInt-
The lateral area of a cone with radius 4 cm is 60 cm2a) Determine the sLant height of the cone, to the nearest centimetreb) Determine the height of the cone, to the nearest centimetre.
The height of a cone is doubled. Does this double the surface area?Justih your answer.
7. The radius of a cone is doubled. Does this double the surface area?Justify your answer.
a) What are the dimensions of the largest cone that fits inside this box?b) What is the surface area of this cone, to the nearest square
centimetre?
9. A cone just fits inside a cylinder. The volume of the cylinder is9425 cm3. What is the surface area of this cone, to the nearest squarecentimetre?
10. The frustum of a cone is the part that remains after the top portionhas been removed by making a cut parallel to the base. Calculate thesurface area of this frustum, to the nearest square metre,
4. One cone has base radius 4 cm and height 6 cm. Another cone hasa base radius 6 cm and height 4 cm.a) Do the cones have the same slant height?b) Do the cones have the same surface area? If not, predict which
cone has the greater surface area. Explain your reasoning.c) Determine the surface area of each cone to check your prediction.
Were you correct?
5.
6.
8. A cube-shaped box has sides 10 cm in length.
-7”
F
20cm
m
448 MHR ‘Chapters
12.
cIr.Dter Problem Emily has obtained an unfinished ceramic birdbath
for one of her customers. She plans to paint it with a special glaze so
that it will he weatherproof. The birdbath is constructed of two parts:
a shallow open-topped cylinder with an outside diameter of 1 m
and a depth of 5 cm, with I-cm-thick walls and base
a ( onical frustum on which the G3t1 inder sits
,‘- 20 cm
10cm
-H 60cm1
Identify the surfaces that are to be painted and describe how
to calculate the area
CaLculate the surface area to be painted, to the nearest square
centimetre.
One can of glaze covers 1 in2. How many cans of glaze will Emily
need to cover all surfaces of the birdbath and the frustum?
Create a problem involving the surface area of a cone, Solve
the problem. Exchange with a classmate.
Extend13.
14.
Suppose the cube in question B has sides of length x
a) Writeexpressions for the dimensions of the largest
cone that (its inside this box.
b) What is a formula for the surface area of this cone?
a) Find an expression for the slant height of a cone in terms
of its lateral area and its radius,
b)
c)
15. Located in the Azores Islands off the coast of Portugal, Mt. Pico
Volcano stands 2351 m tall. Measure the photo to estimate the radius
of the base of the volcano, and then calculate its lateral surface area,
to the nearest square metre
rn Did You Know?
There are 8000 to 10 000
people of Azorean heritage
living in Ontario.
iT
B
a)
m
b) If the lateral area of a cone is 100 cm2 and its radius is
4 cm, determine its slant height.
8.4 surface Area of a Cone ‘ MHR 449
Communicate Your Understanding
O A cylindrical container and a conical container have the same
radius and height. How are their volumes related? I-low could
you illustrate this relationship for a friend?
O Suppose the height of a cone is doubled, Flow will this affect
the volume?
O Suppose the radius of a cone is doubled. How will this affect
the volume?
Practise
For help with question 1, see Example I
1. Determine the volume of each cone. Round to the nearest cubic unit.
/ \/ \3Ocm
10cm
Wesley uses a cone-shaped funnel to put oil ina car engine. The funnel has a radius of 5.4 cm
and a slant height of 10.2 cm. How much oil can
the Funnel hold, to the nearest tenth of a cubic
centimetre?
a) /1\ b)
/‘./\ cm
\ 2cm/1 m
c) 12 mm d)
cm
‘-‘30mm40cm
a)
For help with questions 2 and 3. soc Example 2.
2. Determine the volume of each cone. Round to the nearest cubic: unit.
b)
//\
1m
3. 54cm
10,2
454 MHR ‘Chapter 8
For help iii ii question 4, see Example 3.
4. A cone-shaped paper cup has a volume of 67 cm3 and a diameter
of 6 cm. What is the height of the paper cup, to the nearest tenth
of a centimetre?
Connect and Apply
5. A cone just fits inside a cvi inder with volume 300 cm3
What is the volume of the cone?
5. Create a problem involving the volume of a cone. Solve it.
Exchange your irohiem with a classmate.
7. A cone has a volume of 150 cm3. What is the volume of a
cylinder that just holds the cone?
8. A cone-shaped storage unit at a highway maintenance
depot holds 4000 m3 of sand. The unit has a base radius
of 15 m,
a) Estimate the height of the storage unit.
b) Calculate the height.
c) How close was your estimate?
9. A cone has a height of 4 cm and a base radius of 3 cm.
Another cone has a height of 3 cm and a base radius of 4 cm.
a) Predict which cone has the greater volume. Explain your
prediction.
b) Calculate We volume of each cone, to the nearest cubic
centimetre. Was your prediction correct?
10. Chapter Problem Refer to question 11 in Section 8.4. Determine
the volume of concrete in Emily’s birdbath. Round your answer
to the nearest cubic centimetre.
11. a) Express the height of a cone in terms of its volume and its radius.
b) If a cone holds I L and its radius is 4 cm, what is its height?
Round your answer to the nearest tenth of a centimetre.
12. A cone-shaped funnel holds 120 mL of water. If the height of the
funnel is 15 cm, determine the radius, rounded to the nearest
tenth of a centimetre.
IA
Li z
EO cm
10cm
60 cm
1cm
Im
8.5 Volume ola ConeS MHR 455
Extend
13. A cone just fits inside a cube with sides that measure 10 cm.
a) What are the dimensions of the largest cone
that fits inside this box?
b) Estimate the ratio of the volume of the cone
to the volume of the cube.
c) Calculate the volume of the cone! to the
nearest cubic centimetre.
d) Calculate the ratio in part b).
e) How close was your estimate?
14. A cone has a height equal to its diameter. If the volume of the
cone is 200 m, determine the height of the cone, to the nearesttenth of a metre.
15. Use Technology Use a graphing calculator, The Ocoineter’sSketchpad®. or a spreadsheet to investigate how the volume of acone is affected when its rad ilis is constant and its height changes.
16. Use Technology A cone has a height of 20 cm.
a) Write an algebraic model for the volume of the cone in termsof the radius.
b) Choose a toul for graphing. Graph the volume of the cone versusthe radius.
c) Describe the relationship using mathematical terms
17. Math Contest A cube has side length 6 cm. A square-based pvranii’has side length 0 cm and height 12 cm. A cone has diameter 0 cm
and height 32 cm, A cylinder has diameter 6 cm and height 6 cm.
Order the figures from the least to the greatest volume. Select thecorrect order
A cube! pyramid, cone, cylinder
B cylinder, cube, cone, pyramid
C cube, cone, cylinder, pyramid
D cone, pyramid, cylinder, cube
E pyramid, cone, cylinder, cube
456 MHR ChapterS
Communicate Your Understanding
Describe how you would determine the amount of leather
required to cover a softball.
O Does doubling the radius of a sphere double the surface
area? Explain your reasoning.
1. Determine the surface area o each sphere. Round to the
nearest square unit.b)
30.2 mm
A ball used to play table tennis has a diameter of 40
a) Estimate the surface area of this ball
b) Calculate the surface area, to the nearest square
millimetre. How close was your estimate?
Far help with question 3, see Example 2.
mm.
3. A sphere has a surface area of 42,5 in2. Find its radius,
to the nearest tenth of a metre.
Connect and Apply
basketball has a diameter of 24.8 cm.
How much leather is required to cover this ball,
to the nearest tenth of a square centimetre?
If the loather costs $28/rn2, what does it cost to
cover the basketball?
S. The diameter of Earth is approximately 12 800 km.
a) Calculate the surface area of Earth, to the nearest square kilometre.
b) What assumptions did you make?
• Practise
For help with questions I and 2, see Example I
a)- 6cm
c)
3m
d)
2.
5.6 m
f//i tN
4. A
a)
b)
8.6 Surface Area of a SphereS MHR 459
6. a) The diameter of Mars is 6800 km. Calculate its
surface area, to the nearest square kilometre.
b) Compare the surface area of Mars to the surface
area of Earth from question 5. Approximately
how many times greater is the surface area of
Earth than the surface area of Mars?
7. Chapter Problem Emily is placing a gazing ball in one
of her customer’s gardens. The ball has a diameter
of 60 cm and will be covered with reflective crystals.
One jar of these crystals covers 1 m2.
a) Estimate the surface area to decide whether one
jar of the crystals will cover the ball.
b) Calculate the surface area, to the nearest square centimetre.
c) Was your estimate reasonable? Explain.
B. The radius of a sphere is 15 cm.
a) Predict how much the surface area increases if the radius
increases by 2 cm.
b) Calculate the change in the surface area, to the nearest
square centimetre.
c) lion’ accurate was your prediction?
9. Use Technology
a) Use a graphing calculator to graph the surface area of a sphere
versus its radius by entering the surface area formula.
b) Describe the relationship.
c) Use the TRACE feature to determine
• the surface area of a sphere with radius 5.35 cm
• the radius of a sphere with surface area 80 cm2
Extend10.
11.
12.
Use Technology
a) Determine an algebraic expression for the radius of a sphere in
terms of its surface area.
b) Use your expression from part a) and a graphing calculator to
graph the relationship between the radius and the surface area.
c) Describe the relationship.
d) Use the graphing calculator to find the radius of a sphere with
surface area 200 cm2.
A spherical balloon is blown up from a diameter of 10 cm to a
diameter of 30 cm. By what factor has its surface area increased?
Explain your reasoning.
Which has the Larger surface area: a sphere of radius r or a cube
with edges of length 2r?
p
460 MHR Chapter 8
r
communicate Your Understanding
Describe how you would determine the volume of a sphere
if you knew its surface area.
o How is the volume of a sphere affected if you double
the radius?
Practise
For lie/p with questions ito 3, see Example 1.
1. Calculate the volume of each sphere- Round to the nearest cubic unit.
2.
b)(32mm
c)
A golf ball has a diameter of 4.3 cm. Calculate its
volume, to the nearest cubic centimetre.
Hailstones thought to be the size of baseballs killed
hundreds of people and cattle in the Moradabad and Beheri
districts of India in 1888. The haiistones had a reported
diameter of 8 cm, What was the volume of each one, to the
nearest cubic centimetre?
For lie/p with question 4, see Example 2.
4. A table tennis ball just fits inside a plastic cube with edges 40 mm.
a) Calculate the volume of the table tennis ball, to the nearest
cubic millimetre.
b) Calculate the volume of the cube.
c) Determine the amount of empty space
a) 2.1 m
3.
rrr
r
8.7 Volume of a Sphere. MHR 465
Connect and ApplyS. The largest lollipop ever made had a diameter of 140,3 cm and
was made for a festival in Gränna, Swoden, on July 27, 2003.
a) If a spherical lollipop with diameter 4 cm has a mass of 50 g,what was the mass of this giant lollipop to the nearest kilogram?
b) Describe any assumptions you have made.
6. Chapter Problem Emily orders a spherical gazing ball for one ofher customers. It is packaged tightly in a cylindrical containerwith a base radius of 30 cm and a height of 60 cm.
a) Calculate the volume of the sphere, to the nearest cubiccentimetre.
b) Calculate the volume of the cylindrical container, to the nearestcubic centimetre.
c) What is the ratio of the volume of the sphere to the volumeof the container?
d) Is this ratio consistent for any sphere that just fits insidea cylinder? Explain your reasoning.
7. Golf balls are stacked three high in a rectangular prismpackage. The diameter of one ball is 4.3 cm. What is theminimum amount of material needed to make the box?
8. A cylindrical silo has a hemispherical top (half of asphere). The cylinder has a height of 20 m and a basediameter of 6.5 m.
a) Estimate the total volume of the silo.
b) Calculate the total volume, to the nearest cubic metre.
c) The silo should be filled to no more than 80% capacityto allow for air circulation. 1-low much grain can be putin the silo?
d) A truck with a bin measuring 7 m by 3 m by 2.5 m delivers grainto the farm. How many truckloads would fill the silo to itsrecommended capacity?
9. The tank of a propane tank truckis in the shape of a cylinder witha hemisphere at both ends. Thetank has a radius of 2 m and a totallength of 10.2 m. Calculate thevolume of the tank, to the nearestcubic metre.
466 MHR’ Chapters
Estimate how many basketballs would fit into your classroom.
Explain your reasoning and estimation lechniques and describe any
assumptions you have made. Compare your answer with that of a
classmate. Are your answers close? If not, whose answer is a more
reasonable estimate and why?
• Achievement Check
1. The T-Ball company is considering packaging two tennis balls that
• are 8.5 cm in diameter in a cylinder or in a square-based prism.
a) What are the dimensions and volumes of the two containers?
b) How much empty space would there be in each container?
c) What factors should Ge T-Ball company consider in choosing
the package design? Justify your choices.
Extend
1 . Estimate and then calculate the radius of a sphere with a volume of
600 cm3.
13. Use Technology Graph V=
vr using a graphing calculator.
a) Use the TRACE feature to determine the volume of a sphere with
a radius of 6.2 cm.
b) Check your answer to question 12 by using the TRACE feature to
approximate the radius of a sphere with a volume of 600 cm3.
14. If the surface area of a sphere is doubled from 250 cm2 to 500 cm2,
by what factor does its volume increase?
15. A sphere just fits inside a cube with sides of length 8 cm.
a) Estimate the ratio of the volume of the sphere to the volume of
the cube.
b) Calculate tho volumes of the sphere and the cube and their ratio.
c) How does your answer compare to your estimate?
16. \\ihicli has the larger volume: a sphere of radius r or a cube with
edges of length 2r?
10.
8 cm
8 cm8 cm
8.7 Volume of a Sphere• MHR 467
1. Determine the perimeter and area of eachright triangle. Round answers to the nearesttenth of a unit or square unit.
8.2 cm
ins cm
b)
2. A 6-rn extension ladder leans against avertical wall with its base 2 m frorn thewall. How high up the wall does the topof the ladder reach? Round to the nearesttenth of a metre.
8.2 Perimeter and Area of Composite Figures,pages 426—43 5
3. Calculate the perimeter and area of eachfigure. Round answers to the nearest tenthof a unit or square unit, if necessary.
a) Sm
4. The diagram shows a running track at a highschool. The track consists of two parallelline segments, with a semicircle at each end.The track is 10 rn wide.
a) Tyler runs on the inner edge of the track.How far does he run in one lap, to thenearest tenth of a metre?
b) Dylan runs on the outer edge. How fardoes he run in one lap, to the neatesttenth of a metre?
c) Find the difference between the distancesrun by Tyler and Dylan.
8.3 Surface Area and Volume of Prisms andPyramids, pages 43 6—443
5. Calculate the surface area of each object.Round answers to the nearest square unit,if necessary.
a)
b) the Great Pyramid of Cheops, with aheight of about 147 rn and a base widthof about 230 rn
A sphere has a radius of 3 cm, lVhat is itsvolume, to the nearest cubic centimetre?
A 330 cm3
B 38cm3
C 113 cm3
D 85cm3
2. What is the area of the figure, to the nearestsquare centimetre?
10 cm
/7cm
3. A circular swimming pool has a diameter of7.5 m. It is filled to a depth of 1.4 m. Whatis the volume of water in the pool. to thenearest litre?
A 61850L
B 247400L
C 23501L
D 47124L
4. A conical pile of road salt is 15 m high andhas a base diameter of 30 it. How muchplastic sheeting is required to cover the pile,to the nearest square metre?
A 414 m2
B 90Gm2
C 707m2
0 090m2
5. What is the length of the unknown sideof the triangle, to the nearest tenth ofa millimetre?
4.2mm
Short Response
Show all steps to your solutions.
6. A candle is in the shape of a square-basedpyramid.
a) How much wax is needed to create thecandle, to the nearest cubic centimetre?
b) How much plastic wrap, to the nearesttenth of a square centimetre, would youneed to completely cover the candle?What assumptions did you make?
7. A rectangular cardboard carton is designedto hold six rolls of paper towel that are28 cm high and 10 cm in dinmeter. Describehow you would calculate the amount ofcardboard required to make this carton.
8. Compare the effects of doubling the radiuson the volume of a cylinder and a sphere.Justify your answer with numericalexamples.
9. Calculate the surface area of the cone thatjust fits inside a cylinder with a base radiusof 8 cm and a height of 10 cm. Round to thenearest square centimetre.
10. Determine the volume of a conical pile ofgrain that is 10 m high with a base diameterof 20 m. Round to the nearest cubic metre.
1Dm
2Dm
1.
[r
10cm
A 43 cm2
B 54cm2
C 62cm2
D 73cm2
8 cm
8cm
5cm
I.’
14
L
10cm
A 2.3mm
B 5.0mm
C 6.1mm
D
8 cm
7.7 mm
472 MHR ‘Chapter 8
EXteLLded Response 12. A rectangular carton holds 12 cylindrical
provide complete solutions,cans flat each contain three tennis balls
like the ones described in question 11.
i. Three tennis balls that measure 8.4cm in
diametur arc stackud in a clindrica1 can f(( N’/(
OOuOflflQb
a) Determine the minimum volume
of the can, to the nearest tenth of a a) How much empty space is in each can
cubic centimetre. of tennis balls, to the nearest tenth of a
b) Calculate the amount of aluminum cubic centimetre?
required to make the can, including b) Draw a diagram to show the dimensions
the top and bottom, Round to the of the carton.
nearest square centimetre. c) I-low much empty space is in the carton
c) The can comes with a plastic lid to be and cans once (he 12 cans are placed in
used once the can is opened. Find the the carton?
amount of plastic required for the lid. d) What is tim minimum amount of
Round to the nearest square centimetre. cardboard necessary to make this carton?
dJ Describe any assumptions you have
made.
Chapter Problem Wrap-Up
You are to design a fountain for the garden of one of Emily’s customers.
• The fountain will have a cylindrical base with a cone on top.
• The cylindrical base will have a diameter of 1 m.
• The fountain is to be made of concrete.
• The entire fountain is to be coated with protective paint.
a) Make a sketch of your design, showing all d) Concrete costs $100/rn3. Each litre of
dimensions. protective paint costs $17.50 and covers
b) How much concrete is needed to make the 5 m2. Find the total cost of the materials
fountain?needed to make the fountain.
c) What is the surface area that needs to be
Chapter 8 Practice Test MHR 473
I
15. Answers will vary. Examples:a) The five triangles funned by two adjacent sides of
PQRST (AABC, ABCO, and so on) are isesceles andcongruent WAS). So. all the acute anglcu in thesetriangles are equal. Then, AABR, ABCS, ACUT.
ADEP, and AEAQ are all congruent (ASA). The
obtuse angles of these triangles are opposite to the
interior angles of PQRST. Thus, these nngles are allequal. AUPT, AEPQ, AAQR, ABRS, and ACST are allcongruent WAS), so the sides of PQRST are all equal.
b) Yes; both ore regular pentagons.
c) By measuring the diagram * 2.7
IABVd) Ratio of areas is 7.1.
n(n — 1)17. a)
—
6. a) 95° b) 90°c) c 145°, d 60°,c 85°,f= 95°d)v=55°,w=50°,x=75°y=70°,z= 110°
7. Answers will vary. Examples:a) The sum of the interior angles is 360°. Opposite
interior angles are equal. .djacert interior angles
are supplementacb) The diagonals bisect each other and bisect the
area of the parallelogram.8. Example: A LC = 9n°, LB 60°.LD = 120°9. 2160°
10. 1511. Answers will vary. Example: Run the fence along
the median from the right vertex of the lot.12. a) hexagon
b) Yes, the sides aro equal, and measuring with aprotractor shows that the interior angles are equal.
c) 120°d) For regular polygons, the measure of the interior
angles increases as the number of sides increases.
16. a)45 b) 66
nb — 3)b) ———— -
Chapter 7 RevIew, pages 408—409
1. a) 110° b) 125°c) w = 75°, x = 105°, y = 135°, z = 30°
2. The exterior angle would be greater than 180°.
3. a) any obtuse triangleb) impossible: third exterior angle would he
greater than 180°c) any acute triangled) impossible; sum of exterior angle would
be less than 360°
Chapter 8
100°
Get Ready, pages 414—417
1. a)9.6md) 13.2cm
2. a) 17,6cmd) 39.3 r:m
b) 26 cme) 90 mb) 32.0 m
3.4. a)
b)b=105°,c= 70°,d05°,e100°,f=80’
C) x = 52°. Y 52°. z = 128°5. a) Example: three 110° angles
b) impossible; sum of the interior angles weuldhe greater than 360°
c) Example: three 100° anglesd) impossible; sum of the exterior angles would
be greater than 360°6. a) 720° b) 2080°
7. a) 108° b) 140°
4Dni
c) 6.3 mm1)35mmc) 219.9mm
4.5.6.7.
a) 38.6 cm2a) 11.34 &a) 52 m224 m3; 9425
30
b) 105.7 c&b) 60.45 cm2b) 2513 cm2
cm1
8.9.0.
0. a)
I
c) 1800°c) 157.5°
20 m
11.
Answers t’il) vary.DE connects the miçlpoints of AB and AC. Therefore,the base and altitude of AADE are half those of AABC,
a) Each median divides the triangle into two triangles.All of these triangles are congruent (SAS). Themedians are equal in lengUl since they are sidesof the congruent triangles.
b) False; any scalene triangle is a counter-example.
12.—i 3. Answers will vary.
6.5 m
Chapter? Test, pages 410—41
I. C
b) 685 m2 c) 050 m19.—il. Answers will van’.
2. B3. B4. 05. B
0.1 Apply the Pythagorean Theorem, pages 418—425
I. a) 10 cm2. a) 15 cm3. a) 24 cm24. a) 4.5 units
Li) 13b) 9.2 mb) 34.1 ni2
b) 2.8 units
c) 6.6 mc) 7.7 rn
d) 8.6 cmd) 7.4 cm
5. 35 cm6. 38 m7. hOrn
c) 5 units
8. 104.56 m
574 MHR Answers
9, 11 stones
10, 64 cm
21 22 23 24Ii) —-+ -
2 2 2 2
c) As you sild right triangles to the spiral pattern, the
2 Number of Triangles
area will increase by —— -; —— —
13. a) This name is appropriate because tltissci of three
who] e numbers satisfies the Pythagorean theorem.
b) Yes.
c) Yes, they are Pythagorean triples. with some
restrictions on the values of in and a.
d) m> a> 0
8.2 Perimeter and Area of Composite Figures, pages 420—435
1. a)52m b)2Ocm c)54m
d) 52 cm e) 24 cm
2. a) 370 mm’ b) 104 m’ C) 30cm’
d) 45cm2 0)322 cm’ f) 174 m2
3. a)62m b)232m’
c) To [inc1 the perimeter:
Slop 1: Use the Pythagorean theorem to determine
the length of the unknown side.
Step 2: Add the dimensions of the outer boundary
to determine lie perimeter.
To [ind tl,e area: Use the formula for he area cIa
trapezoid.
4, a) 1500 cm’ b) 1 paint can C) $4.54
5. 300 nun’
6.—i. Answers will vary.
8. a) 180 plants b) 48 m’
9. Answers will vary.
10. a) 1610 cm2 b) 36%
11. a)2.2m b)9m
12. 5400 cm’
14. a)SOm’b) The area of the lawn is four times the area of
one flower bed.
c) It is only true if the vertices of the inscribed
square are at the midpoints.
15. Doobling the radius quadruples the ama. Area,
Area. = ,r(2r)’ or 4,u2. So, Area1 4 X Area,.
16. a) 34, 55, 80, 144
b) areas: 1.2,6, 15, 40.104,
c)—d) Answers will vary.
17. 1:518. 40cm’
8.3 Surface Area and Volume of Prisms and Pyramids,
pages 436-443
1. a) 279.65 cm’
2. a) 2000 nlm’
3. a) 700 mm’
4. a) 400 cm’
5. a) 52 m’
6. 30 cm
7. a) 1 604 000 m’
8. 143.5 m
9. Yes
10. a) 30 cmb) There are no irregularities (humps/dimples) on
the surface. Also, tile top nf the juice container
is flat and the container is completely full.
11. a)47 m’ b) 15 cans c) $202.30
12. a) Answers will vary. Example: 80 rn’
b) 92 000 Lc) 020 mm or 151i and 20 mm
13. a) Answers will vary. Example: Double the volume.
b) original prism 240 cm’; new prism 480 cm’
c) Answers will vary. Yes.
d) Yes; doubling the height of a triangular prism
doubles the volume of the prism.
15. The height of the pyramid is three times the height
of the prism.
Volume of pyramid
Volume of prism /n-h
If the two vol umnes are equal, then the height nf the
pyramid must be tires times the height of the prism
because mc and I are the san,e for both.
16. a) 56 Sn’ b) 510550
17. a) SA = 2(1w + wh + ii)
21121 K 2w) + (2w K 211) + (2/ K 2/mfl
= 214/il’ 4w]s + 4/hi
4(2(1w t IC/I ÷ //ill
b) The new volume is eight times the old volume.
18. 48
Volume,,, 1w],
Vnlunie,,,,,V 2] K 2w K 2k
= Ouch
8.4 5urface Area of a Cane, pages 444-450
I. a) 0 m’ b) 1257 cm’ c) 141 cIa’
2. a)13m b)283m’
3. a)lSBcm’
b) Answers will vary. There is nu paper being
o ‘erl ap pod.
4. a)Yes.b) No. The second cone. The slant height is the same
for both, hot in the expression rs. the second cone
has the greater radius.
t) 141 cm’; 249 cm’; yes
5. a)Scm b)3c,n
6. No. Answers will vary. Example: The formula for the
surface area of the cone is SA i,r’ + ,rm-s. When the
height is doubled only the term ,vrs is changed. The
term ,rr’ remains unaltered. Hencu, doubling the height
of a cone does net doulmle the surface aren.
7. No. Answers will vary. Example: The Formula for the
surface area of a cone is SA trr’ + ,rrs. When the
radius is doubled, the term ,r’ will quadruple a,, d the
term ‘cr5 xvii I more than double. Hence, the surlhce area
of the new cone will bo mnre than double the original
colic.
8. a) radius 5 cm. height 10 cm
b)254 f.m2
9. i:toz cm’
10. lsnm’
11. 4Oft
12. a) 2 2; 2 3; 2 4; 2
b) 147 cm’b) 2 ma’
b) 402 cm’b) 10.35 In’
b) 24 m’
b) 115 324 m’
Answers MHR 515
b) 188 m3d) 25 133 cm’b) 2964 cm’
10. a) rSA
b)9 4,7
c) Answers will vary. Example: The relation isincreasing for all values of r greater than 0 (sincethe radius cannot be negative). The growth rate isnon-linear.
17. 0
8.6 Surface Area of a Sphere, pages 457—461
1. a)452 cm2 b) 11461mm2c)28m2 d)99m’
2. a) Answers will vary, about 4800 mm-’b) 5027 mm; Answers will vafl’.
3. 1.8 m4. a) 1932.2 cm2 b) $5.415. a) 514 718 540 jun2
b) Assumption: Earth is a sphere6. a) 145 267 244 luia2
b) approximately 3.5 times greater7. a) Answers will vary Example: 10800cm2. No; two
jars will be required.b) 11 310 cmc) Answers will vary. Example: Yes; whether von
use the approximate value or the exact va’ue, twojars of reflective crystals are required to cover thegazing ball.
8. a) Answers will van. Example: 750 cm2b) 804 cm c) Answers will vary.
9. a) ywmsz
b) The radius must be greater than 0. As the radiusincreases, the surface area also increases in anon-linear pattern.
c) 360cm-’; 2.5cm
11. a) base of the frustum, lateral area of the frustum, top ofthe frustum, outer walls of the cylinder, inner wallsof the cylinder, the thia strip of the cylinder, theouter part of the base of the cylinder, the inner partof the base of the cylinder
b) 34 382 cm2 c) 4 cans12. Answers will vary.
13. a) radius , height x, slant height =
‘cx’ 25b) SA + -‘rX
Lateral Area14. a) s= ——‘ b)s 7.96 cm
15. Answers will vary, about 72 000 000 m216. a)SA = 4’c + 2’cs
b) Graphs will van. Should be a set points alonga straight line.
c) Answers will vary. Example: ft is a linear relation.
83 Volume of a Cone, pages 451—456
1. a) 25 cm’c) 2827 cm’
2. a) 2 m’3. 264.1 cm’4. 7.1 cm5. luocm’6. Answers will vary.7. 450cm!
8. a) Answers will van’. Example: 18 mb) 16.98 mc) Answer will vary.
9. a) Answers will vary. Example: The cone with baseradius of 4 cm has the greater volume. The formulafor the volume of a cone contains two factors ofand only one factor of h. Hence, the volume is moredependent on r than on h.
b) Cone (height 4 cm. base radius 3 cm):Volume = 38 cm!
Cone (height 3cm, base radius 4cm):Volume 50cm’
10. 141 045 cm3
11. a) h=’f b)59.7cm
12. 2.8 cm13. a) radius 5 cm, height 10 cm b) Estimates will vary.
1:4c) 262 cm3 d) 1:3.82 e) Answers will vary.
14. 9.1 m15. Answers will van’. Example: When the radius is
constant, a change in height produces a proportionalchange in volume.
16. a) V =
H
S
c) TIme radius and the surface area must be greater than0. The trend between die two variables is non-linearwith the radius increasing as the surface areaincreases but at a slower rate.
d) 4 cm11. The surface area has increased hy a factor of nine.
4rr4’c(3r)’
= 4’c(Or’)=
12. TIme cube with edge length 2r.
13. a) Answers will vary. Example:
b) surface area of sphere = 100’c;
surface area of cube 600; ,7:6c) Answers will vary.d) 1:1.91 I
516 MHR Answer5
8.7 volume of a Sphere, pages 462—469 9. 1158 cni
1. a) 11 994 cm3 b) 137 258 mm c) 5 m3
2. 42 cm’3. 268 cm34. a) 33 510 mm3 b) 64 000 mm1 c) 30 490 alm1
5. a) 70.16cm
b) Answers may vary. Example: The largest lollipop
hod the same mass per cubic centimetre as the
small lollipop.6. a) 113 097 cm3 b) 169 646 cin c) 2:3
d) Yes. When the sphere just flis inside the cylinder,
Ii = 2r. So,
about 5200 cm3
4Volume.1,.
Volume,.U.1h.
3
4 1=
3 2
2
3
7. 258.86 cm2
8. a) Answers will vary. b) 736 ni1
c) 588 i& d) 12 tn,ckleads
9. 111 m3
10. Answers will vary.
12. Estimates will vary. Actual radius is 5.23 cm.
13. a) 098.3 cml
b) 5.2cm14. by a factor of about 2.83
15. a) Estimates will var. Example: 1:2
b) Volume of the sphere 268 cm;
Volume of the cube 512 cni’; ,i:6
c) Answers will vary.
16. the cube17. Answers will vary.
18. B19. 365.88 cm3
Review. pages 470—471
1. a) perimeter 32.0 cm; area 411 cm2
b) perimeter 28.4 un; area 31.2 cm-
2. 5,7m
3. a) perimeter 28 m; area 48 m2
b) perimeter 32.6 cm; area 61.8 cm1
4. a) 401.1 In b) 463.9 m c) 62.8 In
5. a) 220 cm2 b) 138 736 ni1
6. a) 6 510 080 cI,12
b) 256 (124 cm2
c) Answers lvii I vary. Example: The side walls of
the tent are flat.
d) Answers lviii vary. Example: The answer is fairly
reasonable as when electing a tent, you lv1uit the
side walls to be as flat and stretr lied as possihile.
7. 9.9cm
8. 283 cm2
10. 3.1 cm
11. 678 cmh Volume1 x Vnlume,;,.
12. 1493.0 on2
13. a) 257 359 270 lan2
b) Earth is a sphere.
c) Answers will vary. Example: about25
14. 5806.5 cm3
15. a) Answers will vary. Example:
b) 5283.07 cm
c) Answers will vary.
Practise Test, pages 472473
1. C2. A3. A4. D5. B6. a) 213 cm3 of wax
b) 236.3 cm2; Assum p1 ion: No plastic cover is being
overlapped.
7. Answers will vary. Example: 5080 cm— if the palier
towels are stacked in three columns with two rolls
in each column.
8. Doubling the radius of a sphere will im rease
(lie volume eight times. Doubling the radius
of a cylinder will quadruple the volume.
9. 523 cm2
10. 1047 m3
11. a) 1396.5 cm:I b) 776 cm1 c) 55 cm2
d) Answers will var’. Example: The circular lid covers
the top of the cylitidrir al din with no side parts.
