Box Problem - Polynomials Review

Box Problem - Polynomials Review

Take out a piece of paper and write down the problem on the next slide.

An open box is to be made from a 9 in. by 10 in. piece of material by cutting equal squares from the corners and turning up the sides. What size square would you cut out if the volume of the box must be 60 cubic inches? 

On your paper, draw a picture of what you think this scenario looks like.

Equal squares were cut out of each corner. We don't know the size of the squares, so we'll label their length "x".

This started out as a 9 in. by 10 in. piece of material before I cut out the corners.

Each square that I cut out has dimensions x by x.

When you fold up each flap, an open box is created!

Multiple Choice

What is the formula for volume?

• $V=l\cdot w\cdot h$

• $V=l\cdot w$

• $V=2l+2w$

In order to write the volume for this box, we must figure out the length, width and height in terms of x.

Multiple Choice

Take a guess at which one you think would represent the length of the open box. It's okay if your guess is wrong.

• L = 10-x²

• L = 10-x

• L = 2x-10

• L = 10-2x

The length used to be 10 inches before each flap was folded up.

Multiple Choice

What is the length of the open box?

• L = 10-x²

• L = 10-x

• L = 2x-10

• L = 10-2x

Multiple Choice

What do you think is the width of the open box?

• W = 9-x²

• W = 2x-9

• W = 9-2x

• W = 9-x

Finding the width in terms of x is the same process as finding the length. The width used to be 9 inches but we folded up each flap, losing two x's.

Multiple Choice

What do you think is the height of the open box in terms of x?

• x

• x²

• 2x

• x-9

• x-10

The height was created from each flap being folded up. The height must be x.

Multiple Choice

Select the volume of the open box in terms of x.

• V(x)=(10-2x)(9-2x)(x)

• V(x)=(10-2x)(9-2x)(x²)

• V(x)=(10-x)(9-x)(x)

• V(x)=(2x-10)(2x-9)(x)

Multiple Choice

The answer you selected should have been V(x)=(10-2x)(9-2x)(x).

Now convert this to standard from by distributing everything.

• $V\left(x\right)=x^2-19x+90$

• $V\left(x\right)=x^3-19x^2+90x$

• $V\left(x\right)=4x^2-38x+90$

• $V\left(x\right)=4x^3-38x^2+90x$

Remember the problem:

An open box is to be made from a 9 in. by 10 in. piece of material by cutting equal squares from the corners and turning up the sides. What size square would you cut out if the volume of the box must be 60 cubic inches?

We will now figure out the size of the squares that we must cut in order to create a volume of 60 cubic inches.

Multiple Choice

How do you think we will figure out the size of the squares?

• Set the function V(x) equal to 0 and solve for x

• Set the function V(x) equal to 60 and solve for x

• Set x equal to 0

• Set x equal to 60

Remember each square cut out from the corners has a side length of x, so we must solve for x.

 $60=4x^3-38x^2+90x$  

Set V(x)=60.

When we solve for x, the equation must equal 0.

• Now we will factor

   $4x^3-38x^2+90x-60$  

Multiple Choice

The first step in factoring anything is to do what?

• Take out the common factor

• Multiply the first and last terms

• List all possible rational roots

Multiple Choice

What is the common factor in

 $4x^3-38x^2+90x-60$  ?

• 2x

• x

• 2

• 4

Multiple Choice

Take out the common factor. What is the result?

 $4x^3-38x^2+90x-60$  

•  $4\left(x^3-9x^2+23x-15\right)$  

•  $2x\left(2x^2-19x+45-30\right)$  

•  $2\left(2x^2-19x+45x-30\right)$  

•  $2\left(2x^3-19x^2+45x-30\right)$  

Multiple Choice

After taking out 2, you should have

 $2\left(2x^3-19x^2+45x-30\right)$  .

We will now use the Rational Root Theorem to continue factoring  $2x^3-19x^2+45x-30$   since grouping doesn't work.


List all factors of 30.

• ±1,2,3,5,6,10,15,30

• ±1,2,3,10,15,30

• ±1,2,3,4,5,6,8,10,15,30

• ±1,2,5,6,15,30

Multiple Choice

 $2x^3-19x^2+45x-30$ 


List all factors of 2.

• ±1,2

• ±1

• ±2

• 1,2

Multiple Choice

 $2x^3-19x^2+45x-30$  

List all possible rational roots by dividing all factors of 30 by all factors of 2.

• ±1,2,3,5,6,10,15,30

• ±1,2,3,5,6,10,15,30, 1/2, 3/2, 5/2, 15/2

• ±1,2,3,5,6,10,15,30, 1/2, 3/2

• ±1,2,3,5,6,10,15,30, 1/2, 5/2

Multiple Choice

 $2x^3-19x^2+45x-30$  
Guess and check your possible rational roots until you get a remainder of 0.
Which of the following results in a remainder of 0?

• x-6

• x-5

• x-3

• x-2

The factored form is

 $0=2\left(x-2\right)\left(2x^2-15x+15\right)$  

 $2x^2-15x+15$   is not factorable so we don't do anything to it

Solving x-2=0, we find that x=2.

Multiple Select

What is x=2 in relation to the problem? Select all that apply.

• The length of the open box with a volume of 60 in³

• The side length of the square cut out of each corner

• The height of the open box with a volume of 60 in³

• The width of the open box with a volume of 60 in³

• The number of squares cut out of the material

We found that in order to have an open box with a volume of 60 in³, the squares must be 2 in. by 2 in.

This is the graph of V(x)=4x³-38x²+90x.

• You can see when the square has a side length of 2 (x=2) the volume is 60 cubic inches (y=60).

What do you think is the biggest volume we can create from this 10 in. by 9 in. piece of material?

Use the graph for help.

Multiple Choice

What is the biggest box we can make?

• One with a volume of 4.757 cubic inches

• One with a volume of 4.5 cubic inches

• One with a volume of 63.114 cubic inches

• One with a volume of 1.577 cubic inches

• One with a volume of 60 cubic inches

The highest point on the graph represents the size of the square needed to be cut out in order to have the greatest volume.

Side length of 1.577 inches, volume of 63.114 cubic inches