l + 2w = perimeter
l + 2w = 24
l = 24-2w
Area = l times w
(24-2w)w = Area.
24w -2w^2 = Area.
If you are doing this using calculus, you would take the derivative of the area and set it equal to zero to find w.
let the side parallel to the wall by y m
let the other two sides be x m each
so we have y + 2x = 24 or y = 24-2x
area = xy
= -2x^2 + 24x
let's complete the square to find the vertex of this parabola
= -2(x^2 - 12x + 36 - 36)
= -2( (x-6)^2 - 36)
= -2(x-6)^2 + 72
the vertex is (6,72)
So the maximumum area is 72 ^2 , when x = 6 and y = 12
d(area) = -4x + 24 = 0 for a max area
x = 6
then y = 24 - 12 = 12
max area = 6(12) = 72
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