Expert: Ahmed Salami - 10/25/2010

Question
A company has set aside $4000 to fence a rectangular portion of land adjacent to their building using the building as one side of the enclosed area. The cost of the fence running parallel to the building is $6 per foot installed, and the fencing for the remaining two sides will cost $4 per foot installed. Find the dimensions of the maximum enclosed area.

Answer
Hi Caleb,
Let the two sides have length x and the parallel side have length y (in feet).
The parallel side would cost $6y and the two sides $(2x).4 = $8x
So,
8x + 6y = 4000
y = (4000 - 8x)/6 = (2000 - 4x)/3
The area of the enclosed space is
A = xy
= x(2000 - 4x)/3
= (2000x - 4x²)/3
The area is maximum when dA/dx = 0, but
dA/dx = (2000 - 8x)/3
equating to zero,
2000 - 8x = 0
8x = 2000
x = 250 ft

y = (2000 - 4x)/3
= (2000 - 1000)/3
= (1000/3) ft

Regards