Question

Arnold is trying to make packing boxes out of 6 foot by 6 foot pieces of cardboard from the local grocer. Help Arnold decide how much he should cut from each corner of the cardboard to make an open box for moving.
Part I: Determining Dimensions
Arnold has been given a 6 foot by 6 foot sheet of cardboard to make an open box by cutting an equal size square from each corner, folding up the resulting flaps, and taping at the corners. Your task is to label dimensions on a sketch with the same size variable cut from each corner.

*You don't have to draw one, just explain what it would look like*










Part II: Analyze
How does each variable expression relate to the length, width, and height of the box when folded?














Part III: Extend your Findings
a. Based upon the variables you used in Part II, write a product for the volume.







b. Expand the product to write a volume function.








c. What domain makes sense for the volume?








d. Guess and check values to find the size cut that produces a maximum volume.
*Six guesses are required*

Answer 1

Part I: Determining Dimensions

Arnold has been given a 6 foot by 6 foot sheet of cardboard to make an open box by cutting an equal size square from each corner, folding up the resulting flaps, and taping at the corners. Your task is to label dimensions on a sketch with the same size variable cut from each corner.

*You don't have to draw one, just explain what it would look like*


Answer:

Base of the box:

     it is a square
     side of the base = 6 foot - x - x. = 6 - 2x

Height of the box: x

Part II: Analyze

How does each variable expression relate to the length, width, and height of the box when folded?

Answer:

length = width = 6 - 2x
height = x


Part III: Extend your Findings
a. Based upon the variables you used in Part II, write a product for the volume.

Answer:

Volume = area of the base × height

Volume = (6 - 2x)² x

b. Expand the product to write a volume function.

Answer:

Volume = (36 - 24x + 4x²)x

Volume = 36x - 24x² + 4x³


c. What domain makes sense for the volume?

Answer:

Since x is a physical dimension x is greater than 0

Since the lenght of the cardboarc sheet is 6 and two squares are cut off, x has to be less than 3

So, the domain is (0, 3)


d. Guess and check values to find the size cut that produces a maximum volume.
*Six guesses are required*

Answer:

x              Volume = 36x - 24x² + 4x³

0.1           36(0.1) - 24(0.1)² + 4(0.1)³ = 3.36
0.5           36(0.5) - 24(0.5)² + 4(0.5)³ = 12.5
1.0           36 - 24 + 4 = 16
1.5           36(1.5) - 24(1.5)² + 4(1.5)³ = 13.5
2.0           36(2) - 24(2)² + 4(2)³ = 8
1.7           36(1.7) - 24(1.7)² + 4 (1.7)³ = 11.49
1.2           36(1.2) - 24(1.2)² + 4(1.2)³ = 15.55

Then you can guess that the maximum volume is pretty close to 16 and it is whenx is close to 1.
2.9