1. 3.6 Direct and Inverse Variation
©2001 by R. Villar
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2. Variation or proportion is used to describe relationships between variable quantities
Most formulas in various science fields can be described using variation.

3. 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

4. When the width is 4 inches, the area is 12 square inches.
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.
a) Write an equation that relates A and W.
b) 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

5. 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

6. Find the volume of the air after the pressure is applied.
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

7. 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

8. 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 I Joint variation model.
600 = k (24)(5)2 Substitute to solve for k..
k =
k = 1 so P = R I2
Find R when P is 200 and I is 1.7.
200 = R(1.7) R =
R = 69 ohms