What is the largest rectangular area that can be enclosed with 400 feet of fencing?

Let the length = x
2 lengths are 2x.
Then you have 400 - 2x for both widths, so the width is 200 - x.
The are if the rectangle is

y = x(200 - x)

y = 200x - x^2

y = -x^2 + 200x

Take the first derivative ans set equal to zero to find a maximum value.

y' = -2x + 200

-2x + 200 = 0

-2x = -200

x = 100

Since the side of the rectangle is 100, all sides measure 100 ft, and you have a square.
The maximum area is 100 ft * 100 ft = 10,000 ft.