2008 Excellence in Mathematics Contest
Team Project Level I
(Precalculus and above)
School Name:
Group Members:
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Reference Sheet
Formulas and Facts
You may need to use some of the following formulas and facts in working through this project. You may not need
to use every formula or each fact.
A = bh
Area of a rectangle
C = 2l + 2 w
Perimeter of a rectangle
A = π r2
Area of a circle
Circumference of a circle
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A = bh
2
Area of a triangle
12 inches = 1 foot
5280 feet = 1 mile
3 feet = 1 yard
16 ounces = 1 pound
2.54 centimeters = 1 inch
100¢ = $1
1 kilogram = 2.2 pounds
1 ton = 2000 pounds
1 gigabyte = 1000 megabytes
1 mile = 1609 meters
1 gallon = 3.8 liters
1 square mile = 640 acres
1 sq. yd. = 9 sq. ft
1 cu. ft. of water = 7.48 gallons
1 ml = 1 cu. cm.
C = 2π r
V = π r 2h
Volume of cylinder
Lateral SA = 2π ⋅ r ⋅ h
Lateral surface area of cylinder
V = lwh
Volume of rectangular prism
− b ± b 2 − 4ac
2a
Quadratic Formula
x=
y2 − y1
x2 − x1
Slope
m=
4
V = π r3
3
Volume of a sphere
tan θ =
sin θ
cos θ
References: inventors.about.com
gono.com
thebullandbear.com
heritage.coca-cola.com
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Coca-Cola - TEAM PROJECT Level I
2008 Excellence in Mathematics Contest
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The Team Project is a group activity in which the students are presented an open ended, problem situation
relating to a specific theme. The team members are to solve the problems and write a narrative about the theme
which answers all the mathematical questions posed. Teams are graded on accuracy of mathematical content,
clarity of explanations, and creativity in their narrative.
Undoubtedly, you all have enjoyed a "delicious and refreshing" can of CocaCola in your life. I must say that it is definitely our pleasure to provide you
with our "second to none" cola beverage. But, did you know what a difficult
task it was to get America's number one selling beverage into a can? And
now some members of our board of directors want me to pursue a change in
the design of the soda can in order to maximize our profits. The
information that I am about to provide you is strictly confidential. If it got
into the hands of one of our competitors, our business future could be in
great jeopardy. Your sensitive handling of this information is imperative.
Allow me to fill you in on the history of Coca-Cola and the soda can before
explaining the current situation.
Originally sold as a fountain drink in America's drug stores, Coca-Cola was invented in May, 1886, by Dr. John S.
Pemberton standing over an old brass or iron kettle in Atlanta, Georgia. He tried over 300 combinations of
different herbs and spices during a period of over four years before he finally came up with a liquid that tasted in
his words "all right". That magic brew was named Coca-Cola, and speculation still exists as to whether the potion
was developed as a soft drink, or as a "cure-all" headache tonic. In fact, up until 1903 Coca-Cola contained trace
amounts of cocaine. It was in that year that the company stopped advertising the drink as a cure for headaches and
other ills.
The name "Coca-Cola" was suggested by Dr. Pemberton's bookkeeper, Frank Robinson. He penned the name
Coca-Cola in the flowing script that is famous today. Coca-Cola was chosen because both words actually named
two ingredients found in the syrup. Robinson also thought that two "C's" would look well in advertising. Willis
Venable first sold Coca-Cola at a soda fountain in Jacob's Pharmacy in Atlanta. During its first year, sales of CocaCola averaged six drinks a day -- adding up to total sales for that year of $50. Since the year's expenses were just
over $70, Dr. Pemberton took a loss. Today, Coca-Cola products are consumed at the rate of more than 834
million drinks per day.
It didn't take long for the taste of Coca-Cola to catch the fancy of Americans and it soon became
available in grocery stores and vending machines. In addition to the drug store fountain, the
beverage became available in glass bottles. Through a misunderstanding by one of the early glass
suppliers for Coca-Cola, the new bottle took its peculiar shape from the seedpod of the Cacao tree-the source of chocolate.
In the late 1930's, beverage companies began experimenting with bottling their drinks in metal cans. To avoid
leakage and absorption, companies lined the can with wax. But many consumers complained that the beverage not
only tasted like Coke, but it also tasted like steel! While improvements in can design took place over the years, it
was said that, "you may be able to put Coke in a can but what comes out isn't Coca-Cola. It's a soft drink, nontoxic, but with a flavor that's as far removed from Coca-Cola as ginger ale is from India Ale" (Business Week,
February 12, 1955). Besides improvements in can design in order to maintain taste and freshness, there were many
experimental can shapes. First, the "cone top can" was used, then the more traditional cylindrical design and
finally, the current design of the recyclable aluminum can.
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That brings us to my current dilemma. Our new Director of Finance, who once took a calculus class, said that,
"the design of our current 12-ounce (355-ml) aluminum soft-drink container is inefficient. It does not have the
right dimensions! It uses too much aluminum!"
I have been ordered to look into this allegation and I naturally look to you for your expert advice. What on earth
could this board member mean that our beverage cans do not have the "right" dimensions? What would the "right"
dimensions be? Do you think that we should change the dimensions of our cans so that is uses less aluminum?
Your accurate and thorough reply is greatly appreciated. If all goes well, a lifetime supply of Coke products is in
your future!
Sincerely,
NOTE: For this project, we will assume that the can should be perfectly cylindrical in shape. We acknowledge
that the current shape is not a perfect cylinder, but will assume so for this project. The dimensions of a can as
currently constructed are shown below.
12 cm
6 cm
First, answer the question assuming that the thickness of the aluminum is the same all around. Then, re-work by
taking into consideration the possibility that the top of the can is thicker than the sides and bottom. It's reasonable
to estimate that the top of a can is three times as thick as the sides and bottoms.
You must use both the first and second derivative test to solve the problem and justify your work.
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GRADING RUBRIC FOR Coca-Cola Project
High School Mathematics Contest - 2008
WORK/RESULTS
•
Accurately reports dimensions in both situations.
____/10
•
Correctly constructs formulas for surface area as a function of radius.
____/10
•
Uses at least 2 different representations to display and confirm solutions
(graphical, numerical (tables), symbolic, verbal)
____/10
•
Properly uses and explains the idea of the first derivative to solve the problem.
____/10
•
Properly employs the second derivative to justify the solution to the problem.
____/10
•
All "work" is adequately explained and desired results are reached.
o Explain what you did and why with any major mathematical steps.
o Simple computations or arithmetic do not need explanation.
o If a solution is shown graphically, explain how the graph solves the problem
o Final solutions include necessary units and interpretation.
____/10
TOTAL
____/60
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