Intermediate Algebra (11th edition)

The inverse variation equation defines a rational function. Another example of

inverse variation comes from the distance formula.

Distance formula

Divide each side by r.

Here, t(time) varies inversely as r(rate or speed), with d(distance) serving as the

constant of variation. For example, if the distance between Chicago and Des Moines

is 300 mi, then

and the values of rand tmight be any of the following.

r=75, t= 4 r= 20, t= 15

r=60, t= 5 r= 25, t= 12

r=50, t= 6 r= 30, t= 10

t=

300

r

,

t=

d

r

d=rt

410 CHAPTER 7 Rational Expressions and Functions

NOW TRY
EXERCISE 4
For a constant area, the height
of a triangle varies inversely
as the base. If the height is
7 cm when the base is 8 cm,
find the height when the base
is 14 cm.

As rincreases, tdecreases.

As rdecreases, tincreases.

If we increasethe rate (speed) at which we drive, time decreases.If we decreasethe

rate (speed) at which we drive, time increases.

Solving an Inverse Variation Problem

In the manufacture of a certain medical syringe, the cost of producing the syringe

varies inversely as the number produced. If 10,000 syringes are produced, the cost is

$2 per syringe. Find the cost per syringe of producing 25,000 syringes.

Let

and

Here, as production increases, cost decreases, and as production decreases, cost in-

creases. We write a variation equation using the variables cand xand the constant k.

cvaries inversely as x.

To find k, we replace cwith 2 and xwith 10,000.

Substitute in the variation equation.

Multiply by 10,000.

Since

Here,. Let.

The cost per syringe to make 25,000 syringes is $0.80. NOW TRY

c= k=20,000 x=25,000

20,000

25,000

= 0.80.

c=

k

x,

20,000= k

2 =

k

10,000

c=

k

x

c= the cost per syringe.

x=the number of syringes produced,

EXAMPLE 4