Question

Mrs. Jones challenged her class with the following problem. Create an open-topped box with the greatest possible volume by cutting congruent squares from each corner of a 8" × 11" piece of construction paper.
1) Express the dimensions of the box in terms of x.
2) Write a formula to represent the volume of the box as a function of x. Write the answer in standard form.
3) Use the function to find the dimensions that will give the greatest possible volume.

Answer 1

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.