DOIN= THE CANCAN
Lyle Frantz Rich Van Gilst Donna Osborn
Class: PreCalculus, Calculus
Materials: Different size cans  one per group or one per person, rulers.
Goals: o To determine the minimum surface area of a can with a given volume
o To find an economical way to package 12 of these cans
Time
Required: Part 1 needs 1015 minutes on two separate days to discuss the project and then several days for each individual to complete the project. Part 2 could be done the same way. If the entire project is to be done in groups, then more class time will need to be allocated.
Background: Find the minimum value for a nonlinear function by either using a graphing calculator or calculus. Find an expression for the minimum surface area using a given volume.
Setting: You are hired by a canning company to try to find ways to save money.
Problem: 1. Using a can with a given amount of a product, find the dimensions of the can which will require the least amount of material.
2. Find the best way to package 12 cans.
Evaluation: In both parts the points should be divided among the assumptions, the restatement of the problem, the equations and calculations used, and the recommendations made.
Extensions: o If the dimensions of the can should be changed to minimize the surface area, then write a letter to the company with calculations, and an explanation of your costsaving idea.
o Generalize the relationship between radius and height of any can with a minimum surface area.
o Try to find the best way to package 24 cans.
Teacher Notes: The two parts of this project should be done separately. After the first part is graded and returned, then the second part can be done.
SAMPLE SOLUTION PART I
The volume of my can is 355 ml which I changed to 344 cubic centimeters. I did the rest of my measurements in centimeters.
Volume = Π r^{2}h
355 = Π r^{2}h
h = 355 / Π r^{2}
Surface Area = 2 Π r^{2} + 2 Π rh
So by substituting for h in this equation I got
Surface Area = 2 Π r^{2} + 2 Π r (355 / Π r^{2}) = 2 Π r^{2} + 710/r
By graphing on the TI82 I found that the minimum value occurs at r=3.84 cm when the surface area is 277.54 square cm and the height is 7.67 cm. Using calculus, I took the derivative of the surface area SA= = 4 Π r  710 / r^{2}. Then setting the equation equal to zero to find the minimum, I found that r = (710 / 4 Π) ^ (1 / 3) which is approximately 3.84 cm. The height will be 7.67 cm.
I recommend that the can=s dimensions should be changed to have radius 3.84 cm and height 7.67 cm.
SAMPLE SOLUTION PART II
I found many different ways to package 12 cans but the only arrangements which gave different surface areas are
DIMENSIONS BY NUMBER OF CANS 
SURFACE AREA (SQ. CM.)

1 X 1 X 12 
2944.84 
1 X 2 X 6 
2355.87 
1 X 3 X 4 
2238.08 
2 X 2 X 3 
1884.70 
I recommend that the 12 soda cans are packaged 2 X 2 X 3 because that uses the least amount of cardboard. This size package is also easy to carry.
_______________________________
THE CANCAN
CANNING
COMPANY
_______________________________
MEMORANDUM
_______________________________
TO: New Employee
FROM: Mr. Think I. Can
Manager CanCan Co.
RE: First Task
Congratulations on your new job at our company. Your first job is to analyze the size of the cans used in selling different items. I am sure that money could be saved if the dimensions of the cans were changed. The amount of the product contained in the can is to stay the same. Only the dimensions of the can are allowed to change. Your job is:
o Use your vast knowledge of mathematics to determine the dimensions of the can which require the least amount of material (aluminum, metal, or tin).
o Compare your calculations to the actual size of the can.
o Write me a letter in which you include:
o Any assumptions made.
o A restatement of the problem for your can and its dimensions.
o Any recommendations for change in the dimensions of the can.
o Any possible reasons for keeping the dimensions of the can the same.
_______________________________
THE CANCAN
CANNING
COMPANY
_______________________________
MEMORANDUM
_______________________________
TO: Employee
FROM: Mr. Think I. Can
Manager CanCan Co.
RE: Second Job
CC:
Your report on the best size can was excellent. Thank you for your excellent effort in your first job for our company. Your next job requires extensive use of your superior algebraic skills. I have decided to accept your decision on the best dimensions for your can. Your product will be packaged in cases of 12 cans. Your job is to:
o Identify a variety of possible ways that 12 cans could be packaged. (Include at least 3 ways).
o Calculate the amount of cardboard needed to make all of your different sized boxes.
o Write a letter to me. In this letter give
o Any assumptions made.
o A restatement of the problem for your can and its dimensions.
o All figures and calculations used.
o Any recommendations for the dimensions of the box needed to package 12 cans.
o Any possible reasons to package your product in a different way.
Funded in part by the National Science Foundation and Indiana University 1995