Page 23 - How To Solve Word Problems In Calculus

EXAMPLE 7
A 1-mile racetrack has two semicircular ends connected by
straight lines. Express the area enclosed by the track as a func-
tion of its semicircular radius. Determine its domain.

Solution
Step1
x

r r

r r

x
Step2
The enclosed area consists of a rectangle whose dimen-
sions are x and 2r and two semicircles of radius r whose com-
2
bined area is πr .
A = 2rx + πr 2

Step3
The perimeter of the figure is the length of the two
straight sides added to the lengths of the two semicircular arcs.
Thus

2x + 2πr = 1 Each semicircular arc has length
πr. Together, they form a com-
2x = 1 − 2πr plete circle whose circumference
1 − 2πr is 2πr.
x =
2
We substitute into the equation obtained in step 2.

1 − 2πr
A(r) = 2r + πr 2
2
2
= r − 2πr + πr 2
= r − πr 2

Since r cannot be negative, r ≥ 0. The perimeter of the track

10