Question

A farmer wants to build a fence enclosing a rectangular region bordering a river. If the farmer has 500 feet of fencing, find the maximum area that can be enclosed.

Answer 1

31,250 ft²

Explanation:

Let x represent the length of fence parallel to the river. Then the ends of the rectangle will have dimesion (500-x)/2, and the total area will be ...

... A = x(500-x)/2

This function describes a parabola with zeros at x=0 and x=500. The vertex (maximum) will be located halfway between these, at x=250.

The maximum size pen has area ...

... A = 250(500 -250)/2 = 31,250 . . . . sq ft