Beginning Algebra, 11th Edition

478 CHAPTER 7 Rational Expressions and Applications

Another example of inverse variation comes from the distance formula.

Distance formula

Divide each side by r.

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

serving as the constant of variation. For example, if the distance between two cities

is 300 mi, then

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

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

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

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=dr ,

t=

d

r

d=rt

NOW TRY
EXERCISE 3
If tvaries inversely as r, and
when , find t
when .r= 6

t= 12 r= 3

As rincreases,

tdecreases.

As rdecreases,

tincreases.

Using Inverse Variation

Suppose yvaries inversely as x, and when Find ywhen

Since yvaries inversely as x, there is a constant ksuch that We know that

when so we can find k.

Equation for inverse variation

Substitute the given values.

Multiply by 8. Rewrite as

Since we let and find y.

Therefore, when NOW TRY

Using Inverse Variation

In the manufacturing 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 to produce 25,000 syringes.

Let the number of syringes produced,

and the cost per unit.

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

creases. Since cvaries inversely as x, there is a constant ksuch that the following

holds true.

c=

x=

EXAMPLE 4

x=6, y=4.

y=

24

x

=

24

6

= 4

y=^24 x , x= 6

k= 24 24 =k k=24.

3 =

k

8

y=

k

x

y= 3 x=8,

y=kx.

y= 3 x=8. x=6.

EXAMPLE 3

NOW TRY ANSWER

3. 6

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