Section 2.6

Variation and Applications

Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives

Find equations of direct, inverse, and combined variation given values of the variables.

Solve applied problems involving variation.

Direct Variation

If a situation gives rise to a linear function f(x) = kx, or y = kx, where k is a positive constant, we say that we have direct variation, or that y varies directly as x, or that y is directly proportional to x. The number k is called the variation constant, or constant of proportionality.

Direct Variation

The graph of y = kx, k > 0, always goes through the origin and rises from left to right. As x increases, y increases; that is, the function is increasing on the interval (0,). The constant k is also the slope of the line.

, 0y kx k

Example

Find the variation constant and an equation of variation in which y varies directly as x, and y = 42 when x = 3.

We know that (3, 42) is a solution of y = kx. y = kx

42 = k 3

14 = kThe variation constant 14, is the rate of change of y with respect to x. The equation of variation is y = 14x.

42

3k

Example

Wages. A cashier earns an hourly wage. If the cashier worked 18 hours and earned $168.30, how much will the cashier earn if she works 33 hours?

We can express the amount of money earned as a function of the amount of hours worked. I(h) = kh I(18) = k 18 $168.30 = k 18 $9.35 = k The hourly wage is the variation constant.

Next, we use the equation to find how much the cashier will earn if she works 33 hours. I(33) = $9.35(33) = $308.55

Inverse Variation

If a situation gives rise to a function f(x) = k/x, or y = k/x, where k is a positive constant, we say that we have inverse variation, or that y varies inversely as x, or that y is inversely proportional to x. The number k is called the variation constant, or constant of proportionality.

For the graph y = k/x, k 0, as x increases, y decreases; that is, the function is decreasing on the interval (0, ).

Inverse Variation

For the graph y = k/x, k 0, as x increases, y decreases; that is, the function is decreasing on the interval (0, ).

, 0k

y kx

Example

Find the variation constant and an equation of variation in which y varies inversely as x, and y = 22 when x = 0.4.

The variation constant is 8.8. The equation of variation is y = 8.8/x.

220.4

(0.4)22

8.8

ky

xk

k

k

Example

Road Construction. The time t required to do a job varies inversely as the number of people P who work on the job (assuming that they all work at the same rate). If it takes 180 days for 12 workers to complete a job, how long will it take 15 workers to complete the same job? We can express the amount of time required, in days, as a function of the number of people working.

( )

(12)12

kt P

Pk

t

18012

2160

k

k

Example continued

The equation of variation is t(P) = 2160/P.Next we compute t(15).

It would take 144 days for 15 people to complete the same job.

2160( )

2160(15)

15144

t PP

t

t

Combined Variation

Other kinds of variation:y varies directly as the nth power of x if there is some positive constant k such that .

y varies inversely as the nth power of x if there is some positive constant k such that .

y varies jointly as x and z if there is some positive constant k such that y = kxz.

ny kx

n

ky

x

Example

Find the equation of variation in which y varies directly as the square of x, and y = 12 when x = 2.

Thus y = 3x2.

2

212 2

12 4

3

y kx

k

k

k

Example

Find the equation of variation in which y varies jointly as x and z and inversely as the square of w, and y = 105 when x = 3, z = 20, and w = 2.

Thus

2

2

3 20105

2

xzy k

w

k

105 15

7

k

k

2 2

77 or

xz xzy y

w w

Example

The luminance of a light (E) varies directly with the intensity (I) of the light and inversely with the square distance (D) from the light. At a distance of 10 feet, a light meter reads 3 units for a 50-cd lamp. Find the luminance of a 27-cd lamp at a distance of 9 feet.

Substitute the second set of data into the equation. The lamp gives an luminance reading of 2 units.

2

2

503

106

IE k

Dk

k

2

6 27

92

E

E