Suppose that 32 feet of fencing is available to enclose a rectangular garden., one side of which will be against the side of a house. What dimensions of the garden will guarantee a maximum area?

Answer 1

A = x*y
32 = x+2y

x = 32-2y
A = (32-2y)y = 32y-2y^2
or
y^2-16y = -A/2
you did not say if you are in algebra or calculus so I will do it the algebra way by completing the square
y^2 -16y + (16/2)^2 = -A/2 + 64

(y-8)^2 = -(1/2)(A-128)
y = 8
x = 16
and the area is 128 ft^2

Answer 2

area=LW= but L=32-2w
area=W(32-2W)=32w-2w^2
but roots of the equation are
0=2w(16-w), w=0, or w=16

since this is a parabola, the max/min will occur at 8 (halfway between the roots).
so w=8, L=16