A beach has to enclose a rectangular area, because some endangered species are nesting there. They have 200 feet of rope to rope off the area with. What is the maximum area that they can rope off?

The maximum area enclosed is a square, then the area will be:

side^2 = (perimeter/4) ^2 = (200 feet/4) ^2 = 50^2 feet^2 = 2500 feet^2.

We can prove that this is a square:

Call x one side, then the other side is: (200 - 2x) / 2 = 100 - x

And the area, S = x (100-x) = 100x - x^2

Maximum area = > S' = 0 = 100 - 2x = > 2x = 100 = > x = 100/2 = 50

=> 100 - x = 100 - 50 = 50.

Then, when showed that the two sides are equal to 50 feet, and the shape enclosed is a square with area = 2500 ft^2.