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
 
 the maximum area he can enclose is 200 feet ^ 2

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