A farmer wants to build a rectangular pen adjacent to a straight river. The pen must have an area of 800 square feet. The side of the pen adjacent to the river requires no fencing, while the other three sides do. What is the minimum amount of fencing required to enclose the other three sides of the pen?

Core Concept

Correct. Letting $x$ be the side parallel to the river and $y$ be the other sides, Area $= xy = 800$, so $x = 800/y$. The fencing is $x + 2y = 800/y + 2y$. Taking the derivative yields $P'(y) = -800/y^2 + 2$. Setting $P'(y) = 0$ gives $y = 20$, meaning $x = 40$, and the total fencing is $40 + 2(20) = 80$ feet.