Question

Suppose that 579ft of fencing are used to enclose a corral in the shape of a rectangle with a semicircle whose diameter is a side of a rectangle as the following figure:


Find the dimensions of the corral with maximum area.
x=___ft.
y=___ft.

Answer 1

The dimensions of the corral with maximum area is x = 162.15 ft and y = 81.07 ft

The perimeter (P) of the corral is:

P = y + x + y + π(x/2)

P = 2y + x + πx/2

579 = 2y + x + πx/2

y = (579 - x - πx/2) / 2 = 289.5 - x/2 - πx/4

The area (A) of the coral:

A = xy + π(x/2)²/2

A = xy + πx²/4

A = x[(579 - x - πx/2) / 2] + πx²/8

A = 579x/2 - x²/2 - πx²/4 + πx²/8

A = 579x/2 - x²/2 - πx²/8

The maximum area is at dA/dx = 0

dA/dx = 579/2 - x - πx/4

0 = 579/2 - x - πx/4

x = 162.15 ft

y = (579 - 162.15 - π(162.15)/2) / 2

y = 81.07 ft

The dimensions of the corral with maximum area is x = 162.15 ft and y = 81.07 ft

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