Question

A track and field playing area is in the shape of a rectangle with semicircles at each end. See the figure. The inside perimeter of the track is to be 1100 meters. What should the dimensions of the rectangle be so that the area or the rectangle is maximized?

Answer 1

The dimensions should be the 1100 divided by the sides

Answer 2

The rectangle with maximum area should have dimensions L = 275 meters (the longer sides) W = 550/ π = 175.16 meters (the shorter sides with semicircles)

Perimeter of a rectangle enclosed by semi circles

The rectangle is enclosed by a semi circle on both sides as follows;

Therefore,

perimeter of a semi circle = 1 / 2 (2πr) = πr

The semi circle is two in numbers. Therefore,

perimeter of the semi circles on both sides = 2πr

diameter = w = 2r

perimeter of the semi circles on both sides = πw

Recall

perimeter of a rectangle = 2(l + w)

perimeter of the track = 2l + πw

1100 = 2l + πw

area = lw

we have to maximized the area of the rectangle.

l = 1100 - πw / 2

substitute it in the area

area = (1100 - πw / 2)w

area = 550w  - πw² / 2

Hence,

using

w = - b / 2a

where

a = π / 2

b = 550

w = 550 / π

Hence,

1100 = 2l + π(550 / π)

1100 = 2l + 550

550 = 2l

l = 550 / 2

l = 275 meters

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