A farmer has 264 feet of fencing and wants to build two identical pens for his prize-winning pigs. The pens will be arranged as shown. Determine the dimensions of a pen that will maximize its area.

Question

moonlitfate · Answer

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carson889 · Answer

P = 4y + 3x = 264ft
3x = 264 - 4y
x = 88 - (4y/3)
A = 2*x*y

Optimization problem:

Insert x of perimeter into area formula: 

A = 2*(88-(4y/3))*y 
A = 176y - (8y^2 / 3)
Take derivative of area:
A' = 176 - (48y/9)

Set A' to equal zero as to find the critical point.
0 = 176 - (48y/9)
48y/9 = 176
48y = 1584
y = 33

P = 4y + 3x = 264
264 = 4*33 + 3x
132 = 3x
x = 44

Therefore the dimensions are x = 44 and y = 33.