A track and field playing area is in the shape of a rectangle with semicircles at each end. The inside perimeter of the track is to be 1500 meters. What should the dimensions of the rectangle be so that the area of the rectangle is a maximum?

1 Answer
Zor Shekhtman
Jul 1, 2016

The rectangle with maximum area should have dimensions
L=375 (the longer sides)
W=750π≈238.73 (the shorter sides with semicircles)

Explanation:

Assume the dimensions of a base rectangle are LxW with sides of the length W being diameters of semicircles.

Then the perimeter of a track equals to the lengths of two other sides (2L) plus the length of two semicircles (or, equally, one full circle) of a diameter W, that πW.
The total perimeter is, therefore
(a) 2L+πW=1500

We have to maximize the area of a rectangle, that is
(b) A=L⋅W.

From equation (a) we can derive
(c) L=1500−πW2
and substitute it into the formula (b) for area of rectangle:
A=1500−πW2⋅W=−π2W2+750W

This expression is a quadratic polynomial of W with negative coefficient at W2. As we know, it reaches its maximum at
W=−7502⋅(−π2)=750π

Having W determined, use equation (c) to determine L:
L=1500−π(750π)2=375

CHECK against equation (a):
2⋅375+π750π=750+750=1500

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