Remark
This is going to require that you differentiate an expression. But you have to get the expression first.
Step One
Label the shorter side as 8 - (x + x) = 8 - 2x That should be the length between the 2 xs. That line is the width
Do the same thing for the bottom line. It is 11 - 2x
The height is x
Step Two
Write The givens
W = 8 - 2x
L = 11 - 2x
H = x
Step Three
Write an expression for the Volume and substitute your givens into the volume formula.
V = L * W * H
V = (11 - 2x)(8 - 2x)(x)
Step Four
Expand the Length and Width First. Don't do anything with the Height just yet. Use Foil.
V = (11*8 - 11*2x - 2x*8 + (2x)^2 ) ( x)
V = (88 - 22x - 16x + 4x^2) * x
V = (88 - 38x + 4x^2 )*x Multiply the x by what is inside the brackets.
V = (88x - 38x^2 + 4x^3)
Step Five
Differentiate the Volume
dV/dx = 88 - 76x + 12x^2
The maximum occurs when dV/dx = 0
Step 6
Equate dV/dx to 0 and use the quadratic equation to solve.
12x^2 - 76x + 88 = 0
a = 12
b = - 76
c = 88
I'm going to assume that if you know how to differentiate, you know how to use the quadratic equation. This will give you two answers, both of which will give you the same maximum volume.
x1 = 4.81 or x = 1.53
The first result is not a valid one. Do you see why? The problem is not in the length or in the height. The problem is in the width.
W = 8 - 2x
W = 8 - 2*4.81
W = 8 - 9.62 = - 1.62. You can't have a minus width.
So the only valid answer is x = 1.53
Answer
x = 1.53
Height = 1.53
Width = 8 - 2*1.53 = 4.94
Length = 11 - 2*1.53 = 7.94
Graphs
The graph on the left shows you what the quadratic looks like. It gives a hint of where V cuts the x axis.
The graph on the right shows you what the quadratic looks like as a close up.
