2008 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members:

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1 008 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members:

2 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 C = l + w A = π r Area of a rectangle Perimeter of a rectangle Area of a circle 1 y y C = π r A bh m = x x Circumference of a circle Area of a triangle Slope = inches = 1 foot 580 feet = 1 mile 3 feet = 1 yard 16 ounces = 1 pound.54 centimeters = 1 inch 100 = $1 1 kilogram =. pounds 1 ton = 000 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. 4 3 V = π r h V = lwh V = π r 3 Volume of cylinder Volume of rectangular prism Volume of a sphere Lateral SA = π r h Lateral surface area of cylinder b ± b 4ac x = a Quadratic Formula tan θ = sinθ cosθ References: inventors.about.com gono.com thebullandbear.com heritage.coca-cola.com

3 Coca-Cola - TEAM PROJECT Level I 008 Excellence in Mathematics Contest 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 Coca- Cola 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 Coca- Cola 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 1, 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. 3

4 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 1-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. 1 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. 4

5 GRADING RUBRIC FOR Coca-Cola Project High School Mathematics Contest WORK/RESULTS Accurately reports dimensions in both situations. /10 Correctly constructs formulas for surface area as a function of radius. /10 Uses at least 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. /10 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. TOTAL /60-5

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