Problem 2

Suppose a rancher wants to enclose an area along the side an existing fence. If the enclosed area is to be rectangular and the rancher has a fixed amount of fencing material to make the other three sides, what ratio of length to width will result in the largest possible area?

[Problem submitted by Vin Lee, LACC Associate Professor of Mathematics.]

Solution:

Let p be the total length of the 3 new sides of the rectangle, x the length of the side parallel to the old fence, and y the length of each of the 2 sides perpendicular to the old fence. Then

Area = xy

This is a quadratic equation in standard form. If it is graphed with Area on the vertical axis and y on the horizontal axis, then the graph is a downward opening parabola whose maximum value of Area occurs at the vertex when . So,

.

So, the maximum area occurs when the ratio of length to width is = 2.