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2.4 and10.2 Direct/Inverse/Joint
Variation
©2001 by R. Villar
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Direct, Inverse and Joint Variation
Direct Variation is a linear function.
For example d = r • t
If time is constant, as rate goes up, distance goes up.
We can say, distance varies directly with rate...
Direct variation equations can be written as y = kx where
k is the constant of variation.
Example: The values of x and y vary directly. Write an
equation that relates the variables if x = 3 when y = 12.
y = kx
12 = k • 3
k=4
y = 4x
Example: The area A (in square inches) of a rectangle varies
directly with its width, W, (in inches). When the width is 4 inches,
the area is 12 square inches.
Write an equation that relates A and W.
Find the width of the rectangle when the area is 24 square inches.
W = k•A
This is the model for inverse variation.
4 = k • 12
Substitute to solve for k..
1/3 = k
W = 1/3 • A
Find W when A is 24.
W = 1/3 • 24
W = 8 inches
Inverse Variation
The variables x and y vary inversely if, for a constant k,
yx = k
or
y= k
x
The number k is the constant of variation.
Example: The variables x and y vary inversely. Find an
equation that relates the variables if x = 4 when y = –3.
yx = k
yx = –12
Example: The volume of a gas V at a constant temperature varies
inversely with its pressure, P. The pressure acting on 15 m3 of air is
raised from 1 atmosphere to 2 atmospheres while the temperature
is kept constant.
Find the volume of the air after the pressure is applied.
VP = k
This is the model for inverse variation.
(15)(1) = k
Substitute to solve for k..
15 = k
VP = 15
or
V = 15
P
Find V when P is raised to 2 atmospheres.
V = 15
2
V = 7.5 m3
Joint Variation:
Occurs when a quantity varies directly as the product of
two or more other quantities.
For example, if z = kxy, then z varies jointly with the
product of x and y.
Example: The variable z varies jointly with the product of
x and y. Find an equation that relates the variables if
x = 4 when y = –3 and z = 2.
z = kxy
2 = k(4)(–3)
2 = –12k
k = –1/6
z = –1/6 xy
Example: The Power P in watts of an electrical circuit varies
jointly as the resistance R and the square of the current I.
For a 600-watt microwave oven that draws a current of 5.0 amperes,
the resistance is 24 ohms.
What is the resistance of a 200-watt refrigerator that draws a current
of 1.7 amperes?
P = k R I2
Joint variation model.
600 = k (24)(5)2 Substitute to solve for k..
k = 600
600
k = 1 so P = R I2
Find R when P is 200 and I is 1.7.
200 = R(1.7) 2
R = 200
2.89
R = 69 ohms