A farmer wants to enclose three sides of a rectangular area that borders a creek. He has 2400 meters of fencing material. What is the maximum area that can be enclosed by the fence?

Answer 1

Let L be the length of fence parallel to the river.

A = L*(2400-L)/2 = 1200L - L^2/2
dA/dL = 0 = 1200 - L (using calculus)
L = 1200 m for maximum A
Amax = 1200*600 = 720,000 m^2

You can get the same answer by completing the square.

A = -(1/2)(L^2 -2400L + 1,440,000) + 720,000
= (-1/2)(L-1200)^2 + 720,000

That obviously has a maximum value when L = 1200.