If 800 feet of fencing is available, find the maximum area that can be enclosed.

If 800 feet of fencing is available, find the maximum area that can be enclosed.

Perimeter of a rectangle is:
2l + 2w = P

However, we're given one side (length) is bordered by the river and the fence length is 800, so we have:
So we have l + 2w = 800

Rearranging in terms of l, we have:
l = 800 - 2w

The Area of a rectangle is:
A = lw

Plug in the value for l in the perimeter into this:
A = (800 - 2w)w
A = 800w - 2w^2

Take the first derivative:
A' = 800 - 4w

Now set this equal to 0 for maximum points:
4w = 800

Typing this equation into the search engine, we get:
w = 200

Now plug this into our perimeter equation:
l = 800 - 2(200)
l = 800 - 400
l = 400

The maximum area to be enclosed is;
A = lw
A = 400(200)
A = 80,000 square feet