a woman wants to build a rectangular garden. she plans to use a side of a shed for one side of the garden. she has 84 yd of fencing material.

what is the maximum area that will be enclosed?

The perimeter in this case will be:

p=2x+y  and we are told that p=84yd so

2x+y=84

y=84-2x

Now for the area:

A=xy, and using y found above in this gives you:

A=x(84-2x)

A=84x-2x^2  now taking the derivatives gives you the velocity and acceleration of the area function...

dA/dx=84-4x and d2A/dx2=-4

Since the acceleration is a constant -4, when velocity is equal to zero it is at the absolute maximum value for the area function...

dA/dx=0 only when 84=4x, x=21

So the maximum possible area will occur when x=21

A(21)=84(21)-2(21^2)

A(21)=882 yd^2