Maximum Volumes.....Click Here to Download File
A Tacky Problem
You have a job as a packaging engineer for ABC Office Supplies, a company that manufactures tacks. You are in charge of a design team that wishes to produce a square-based box with surface area of 150 cm2 that has the maximum volume possible.
Problem: What dimensions of a square-based rectangular prism with surface area 150 cm2, will make the greatest volume to hold the greatest number of tacks?
Recall the formulas for Surface Area and Volume
SA = 2(lw + lh + wh) V = lwh
Hypothesis: Make a hypothesis about the dimensions of the box that will give the largest volume.
Data Collection: Calculate the Surface Area and volume of different possible rectangular prisms.
1 a) Consider a prism that has a length of 1 cm and a width of 1 cm and a surface area of 150 cm2.
i)
Calculate the height box.
ii) Calculate the volume of the box.
b) Consider a prism that has a length of 2 cm and a width of 2 cm and a surface area of 150 cm2.
iii)
Calculate the height box.
iv) Calculate the volume of the box.
2. Complete this chart to determine the volumes for other possible rectangular prisms. Use a spreadsheet program if available.
|
Length (cm) |
Width (cm) |
Height (cm) |
Surface Area (cm2) |
Volume (cm3) |
|
1
|
1 |
|
150 |
|
|
1.5
|
1.5 |
|
150 |
|
|
2
|
2 |
|
150 |
|
|
2.5
|
2.5 |
|
150 |
|
|
3
|
3 |
|
150 |
|
|
3.5
|
3.5 |
|
150 |
|
|
4
|
4
|
|
150 |
|
|
4.5
|
4.5 |
|
150 |
|
|
5
|
5 |
|
150 |
|
|
5.5
|
5.5 |
|
150 |
|
|
6
|
6 |
|
150 |
|
|
6.5
|
6.5 |
|
150 |
|
|
7
|
7 |
|
150 |
|
3. Present your data from the table in Question #2 by making a scatter plot of length vs. area. Length is on the horizontal axis and area is on the vertical axis.
Be sure that the graph includes a title and that the axes are labeled.
4.
Draw the line or curve of best fit.