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.