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PAP Algebra 2 NOTES 9.4
TLW…Simplify and work problems dealing with direct, inverse, and joint variations.
K = Constant of VariationDirect
Variation
The equation is:
How to find k:
Inverse Variation
The equation is:
How to find k:
Y=kx x
ky
x
yk xyk
K = Constant of VariationJoint Variation
Joint variation is the same as direct variation with two or more quantities. Example of Joint Variation
y = 7x2z3, here y
varies jointly with x2 and z3 (k=7).
Given: y varies directly with x and y = 12 when x = -4
1. What is the constant
of variation (k) ?
kxy 412 k
k 3
2. Find y when x = 5
xy 3
53y
15y
5
6,2
7
x
ky
275
6 k
2.410
42k
Find k in each Inverse Variation
(5,10) and (2,y) are from the same inverse variation. Find y.
x
ky
510
k
k50
xy50
2
50y
25y
x and y vary inversely where x=-2 & y=1/2 Write a function to model inverse variation.
Find y if x=1/2
x
ky
22
1
k
k 1
xy
1
211
y
2y
If z varies directly with x and inversely with y, the equation for this is:
kz x
y
Suppose that z varies jointly with x and y when x = 2 and y = 3, then z = 60.
A. Write the function that models the variation.
Find the value of “k” first ))(( yxkz
)3)(2(60 kk10
Now you can write the equation
xyz 10B. Find the value of z when x = 4 and y = -10
xyz 10)10)(4(10 z 400z
XX YY
33
-2-2
99
44
Is the relationship between the variables direct or inverse?
6
1
2
1
4
1
2
1
Direct : y = kx so k = y/x
Inverse: y = k/x so k = xy
Find “k” using both equations. If the value of “k” remains the same each time, then it represent that variation.
Neither
18
1
8
1
18
1
8
2
1
2
1
2
9
2
XX YY
11 33
-5-5 -15-15
22 66
1010 3030
Is the relationship between the variables direct or inverse?
Find “k” using both equations. If the value of “k” remains the same each time, then it represent that variation.
Direct
xy 3
Direct : y = kx so k = y/x
Inverse: y = k/x so k = xy
3k
Describe the combined variation that is modeled by the formula:
Molar mass varies directly with density and temperature
essure
eTemperaturDensity
Pr
))((4.62Molar Mass
and inversely with pressure
BTW… k = 62.4
Describe the combined variation that is modeled by the equation:
y varies directly with w2 and √ x
3
2
3
2
z
xwy
and inversely with z3
BTW… k = 2/3
If m varies directly as the square root of y, inversely as p5, and directly as n, what happens to m when y is
multiplied by 9, p is multiplied by 2 and n is quadrupled?:
5p
ynmEquation: m is multiplied by
52
94
p
ynnew
532
34
p
yn 532
12
p
yn
58
3
p
yn
8
3BTW… k = 1
A drama club is planning a bus trip to NYC to see a Broadway play. The cost per person for the bus rental varies inversely as the number of people going on the trip. It will cost $50
per person if 20 people go on the trip. To the nearest dollar, how much will it cost per person if 100 people go on the trip?
p
kc
2050
k
1000k
pc1000
personc /10$
100
1000c
Suppose that m and r vary inversely and that r=4/9 when m=6.Write a function that models the inverse variation
and find r when m=2
r
km
94
6k
9
24k
rm 9
24
r924
2
9
242 r
2
1
9
24r
3
4
18
24r
The amount of oil used by a ship travelling at a uniform speed varies jointly with the distance and the square of the speed. The ship uses 500 barrels of oil in travelling
50mi at 30mph. Write an equation that models this variation and find k. Also find how many barrels of oil
are used when the ship travels 10mi at 45mph.2kdro
23050500 k
k45000500
2
90
10 dr
barrelso 225
2451090
1o
90
1k
20259
1o
Homework #4 Due next class
You next quiz will cover Direct, Inverse and Joint Variation problems