Ramon wants to fence in a rectangular portion of his back yard against the back of his garage for a vegetable garden. he plans to use 40 feet of fence, and needs fence on only three sides. find the maximum area he can enclose. (hint: the lengths of the 3 fenced sides of the rectangle must add up to 40.)

The first thing to do is find the perimeter of the fence:

l + 2w = 40,

l = length

w = width

By definition the area of a rectangle is:

A = l * w

Clearing the perimeter found:

l = 40-2w

We substitute l in the expression of the area:

A = w * (40-2w)

We rewrite:

A = - 2w ^ 2 + 40w

We observe that it is a parabola that opens downwards

We find the maximum of the function:

A ' = - 4w + 40 = 0

4w = 40

w = 10

Substituting in the expression of length:

l = 40-2 (10) = 20

The maximum area is then:

A = (10) * (20) = 200 feet ^ 2

Answer:

the maximum area he can enclose is 200 feet ^ 2