Intermediate Algebra (11th edition)

SECTION 7.6 Variation 409

One variable can be proportional to a power of another variable.

Direct Variation as a Power

yvaries directly as thenth power ofxif there exists a real number ksuch that

ykxn.

The formula for the area of a circle, , is an example. See FIGURE 8. Here,

is the constant of variation, and the area varies directly as the squareof the radius.

a =pr^2 p

r

a = πr^2

FIGURE 8

Solving a Direct Variation Problem

The distance a body falls from rest varies directly as the square of the time it falls

(disregarding air resistance). If a skydiver falls 64 ft in 2 sec, how far will she fall in

8 sec?

Step 1 If drepresents the distance the skydiver falls and tthe time it takes to fall,

then dis a function of tfor some constant k.

dvaries directly as the square of t.

Step 2 To find the value of k, use the fact that the skydiver falls 64 ft in 2 sec.

Variation equation

Let and.

Find k.

Step 3 Now we rewrite the variation equation using 16 for k.

Here,.

Step 4 Let to find the number of feet the skydiver will fall in 8 sec.

Let.

The skydiver will fall 1024 ft in 8 sec. NOW TRY

d= 161822 = 1024 t= 8

t= 8

d= 16 t^2 k= 16

d=kt^2

k= 16

64 = k 1222 d= 64 t= 2

d= kt^2

d=kt^2

EXAMPLE 3

As pressure

on trash

increases,

volume of

trash

decreases.

FIGURE 9

OBJECTIVE 3 Solve inverse variation problems. Another type of variation is

inverse variation.With inverse variation, where ,as one variable increases,

the other variable decreases.

For example, in a closed space, volume decreases as pressure increases, which

can be illustrated by a trash compactor. See FIGURE 9. As the compactor presses down,

the pressure on the trash increases, and in turn, the trash occupies a smaller space.

k> 0

Inverse Variation

yvaries inversely asxif there exists a real number ksuch that

Also, yvaries inversely as thenth power ofxif there exists a real number ksuch

that

y

k

xn

.

y

k

x

.

NOW TRY
EXERCISE 3
Suppose yvaries directly as
the square of x, and
when. Find ywhen
x= 7.

x= 5

y= 200

NOW TRY ANSWER

3. 392