Question

A rancher has 800m of fencing to enclose a rectangular cattle pen along a riverbank. No fencing is required along the riverbank. Find the dimensions that would maximize the area for the cattle to graze. Also, what is the maximum area?

Answer 1

Let the dimensions be L and W. Then P = perimeter = 800 m = 2W + L.

Constraint:

We want to maximize the area. This area is L*W. We can eliminate one variable or the other by solving 800=2W + L for either. L = 800 - 2W. Then

the area is A(W) = (800 - 2W)*W.

Differentiate this with respect to W: A '(W) = (800 - 2W)(1) + W(-2), or

800 - 2W - 2W = 800 - 4W

Set this derivative = to 0 and solve for W: W = 800/4, or W = 200. Then L = 800 - 2W becomes L = 800 - 2(200) = 400.

Thus, W = 200 m and L = 400. From this data we get max area = (200)(400) m^2, or 80000 m^2.