Page 62 - Calc for the Clueless

A Texas rancher has an 800-square-mile rectangular plot of land he wishes to divide into three equal regions, as
pictured below.

A. Find the dimensions so that the rancher uses the least amount of fencing.

B. If it costs 4 times as much per mile to fence the outside, find the dimensions to minimize the cost.

There are two separate problems here. They are done similarly. It is necessary to do them separately, of course.

A. The area in each case equals length times the width. A = xy = 800. So y = 800/x. The fence length f = 2x +
-1
4y = 2x + 4(800/x) = 2x + 3200x .

B. We do not know the cost. So let C equal the cost per mile of the inner fence and 4C the cost per mile of the
outer fence. (C is a constant, which we don't know.) The cost of the fence equals the number of fencing miles
times the cost per mile.

Miles of Fence × Cost per Mile =Total Cost

Outside fence 2x + 2y 4C 4C(2x + 2y)

Inside fence 2y C 2Cy