A farmer has 36 feet of fence to build a pigpen. He is going to use one of the sides of his barn as a side to the rectangular enclosure. Determine a function A that represents the total area of the enclosed region. What is the maximum area that can be enclosed?

Answer 1

A = L * w
Perimeter = L + 2 w = 36
so
L = 36-2w

A = (36-2w)w

-2 w^2 + 36 w = A

w^2 - 18 w = -A/2

w^2 - 18 w + 81 = -A/2 + 81

(w-9)^2 = -(1/2)(A-162)

vertex at
w = 9
area = 162 ft^2
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check
if w = 9
L = 36 - 18 = 18
area = 162, sure enough