Miguel buys 100 feet of fence to enclose a rectangular area of his backyard so his dog can run freely. What is the maximum area, in square feet, he can enclose?

Suppose the length of the triangle is x, if the perimeter of the rectangle is 100 ft, the width of the rectangle will be (50-x) ft.

Area of rectangle will be:

A=length*width

A=x (50-x)

A=50x-x^2

at maximum area, dA/dx=0

thus

dA/dx=50-2x=0

solving for x we get

2x=50

x=25

thus for maximum area length=25 ft

the size of the width will be

50-x=50-25=25 ft

thus the maximum area will be:

25*25=625 sq. feet