Beginning Algebra, 11th Edition

SECTION 7.8 Variation 477

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 sky diver falls 64 ft in 2 sec, how far will she fall in

8 sec?

Step 1 If drepresents the distance the sky diver falls and tthe time it takes to

fall, then dis a function of t, and, for some constant k, the following holds

true.

Step 2 To find the value of k, use the fact

that the sky diver falls 64 ft in 2 sec.

Variation equation

Let and

Apply the exponent.

Divide by 4.

Step 3 Substitute 16 for kto find the varia-

tion equation.

Step 4 Now let to find the number of feet the sky diver will fall in 8 sec.

Let

The sky diver will fall 1024 ft in 8 sec. NOW TRY

OBJECTIVE 2 Solve inverse variation problems.In direct variation, where

as xincreases, yincreases. Similarly, as xdecreases, ydecreases. 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, as illustrated by a trash compactor.

See FIGURE 3. As the compactor presses down, the

pressure on the trash increases. As a result, the trash

occupies a smaller space.

k 7 0,

k 7 0,

d= 161822 = 16 # 64 = 1024 t=8.

t= 8

d= 16 t^2

k= 16

64 = 4 k

64 =k 1222 d= 64 t=2.

d=kt^2

d=kt^2

NOW TRY EXAMPLE 2
EXERCISE 2
If the height is constant, the
volume of a right circular
cylinder varies directly as the
square of the radius of its
base. If the volume is 80
when the radius is 4 ft, find
the volume when the radius
is 5 ft.

ft^3

NOW TRY ANSWER
2.125 ft^3

Inverse Variation

yvaries inversely as xif there exists a real number ksuch that the following is

true.

Also, yvaries inversely as the nth power of xif there exists a real number ksuch

that the following is true.

y

k

xn

y

k

x

As pressure

increases,

volume

decreases.

FIGURE 3