Expert: Paul Klarreich - 2/1/2009

Question
the infield of a 440 yard track consists of a rectangle and 2 semicircles. to what dimensions should the track be built in order to maximize the area of the rectangle?

Answer
Questioner:   amanda
Category:  Calculus
Private:  No
 
Subject:  max-min optimization
Question:  the infield of a 440 yard track consists of a rectangle and 2 semicircles. to what dimensions should the track be built in order to maximize the area of the rectangle?
 
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Hi, Amanda,

I assume you have read the (many) other Maximum-minimum problems, so we can go right away to:

Variables:

2x = width of rectangle.
y = length ............

Then A = 2xy

constraint: Perimeter = 440

Perimeter = 2y + (circle) = 2y + 2 pi x = 440


y + pi x = 220

y = 220 - pi x

Substitute:

A = 2x(220 - pi x)

A = 2(220x - pi x^2)

Diff, set = 0, solve, etc.

I'll leave that to you.