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?

Question

Mathematics

Peyton Boyd

24 April, 02:42

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?

+2

Answers (1)

Know the Answer?

Grace Hanna · 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.