DOIN= THE CAN-CAN
Lyle Frantz
Rich Van Gilst
Donna Osborn
Class:
Pre-Calculus, Calculus
Materials:
Different size cans - one per group or one per person, rulers.
Goals:
o
o
Time
Required:
To determine the minimum surface area of a can with a given volume
To find an economical way to package 12 of these cans
Part 1 needs 10-15 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 non-linear 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
o
o
Teacher Notes:
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 cost-saving idea.
Generalize the relationship between radius and height of any can with a
minimum surface area.
Try to find the best way to package 24 cans.
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.
Funded in part by the National Science Foundation and Indiana University 1995
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 r2h
355 = Π r2h
h = 355 / Π r2
Surface Area = 2 Π r2 + 2 Π rh
So by substituting for h in this equation I got
Surface Area = 2 Π r2 + 2 Π r (355 / Π r2) = 2 Π r2 + 710/r
By graphing on the TI-82 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 / r2. 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
1X2X6
2355.87
1X3X4
2238.08
2X2X3
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.
Funded in part by the National Science Foundation and Indiana University 1995
_______________________________
THE CAN-CAN
CANNING
COMPANY
_______________________________
MEMORANDUM
_______________________________
TO:
New Employee
FROM:
Mr. Think I. Can
Manager Can-Can 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.
Funded in part by the National Science Foundation and Indiana University 1995
_______________________________
THE CAN-CAN
CANNING
COMPANY
_______________________________
MEMORANDUM
_______________________________
TO:
Employee
FROM:
Mr. Think I. Can
Manager Can-Can 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
Funded in part by the National Science Foundation and Indiana University 1995