Among all the rectangles whose perimeters are 100 feet, find the dimensions of the one with the maximum area. explain step by step.

We let x be the length and y be the width of the rectangle. Then,

Perimeter = 2x + 2y

100 = 2x + 2y

50 = x + y

y = 50 - x

Area = xy

A = x (50 - x)

A = 50x - x^2

We then take the derivative; set it equal to zero:

A ' = 50 - 2x

0 = 50 - 2x

2x = 50

x = 25

y = 50 - x

y = 50 - 25

y = 25

Therefore, the dimensions are 25 and 25.