A man wants to construct a rectangular enclosure around his model railroad. Only three sides must be fenced, since his garage wall will...

We want to have the rectangular enclosure with a maximum
area and with three sided fencing of 24 meters.

We assume
that the two parallel sides of the rectangular eclosure measure x meters each and the
remaining third side is 24-2x meters, as the sum of all 3 sides is 24
meters.

Therefore, the area A(x) =  product of two adjacent
sides of the rectangle.

So A(x) = x*(24-2x) , Or A(x) =
24x-2x^2.

A(x) is maximum for some x = c , if A'(c) = 0
and  A" (c) < 0.

Therefore, we differentiate A(x) =
24x-2x^2:

A'(x) = (24x-2x^2)' = 24-4x.  A'(x) = 0 gives 24
-4x = 0 which gives x = c = 24/4 = 6.

Therefore x = c =
6.

A"(x) = (24-4x)' = -4.  So A"(6) = -4 which is <
0.

Therefore A(x) is maximum for  x = c=
6.

So the rectangular enclosure has the maximum area for
the rectangle of width = 6 meters and length = 24-2*6 = 12 meters. Therefore, area =
6*12 = 72 sq meter.