2016 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members:

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1 016 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 r Area of a rectangle Perimeter of a rectangle Area of a circle C r 1 A bh V r dx Circumference of a circle Area of a triangle Volume b a a b c 580 feet = 1 mile V rhdx Pythagorean Theorem b a Volume b dy s 1 dx dx.54 centimeters = 1 inch a Arc Length h 4.9t v0t h0 h 16t v0t h0 1 kilogram =. pounds 1 meter = inches 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 r h V Area of Base height V r 3 Volume of cylinder Volume Volume of a sphere Lateral SA = r h b b 4ac x V a ( R r ) dx Lateral surface area of cylinder Quadratic Formula Volume b a

3 TEAM PROJECT Level I 016 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. We encourage the use of a graphing calculator. Part 1: Background Sources: en.wikipedia.org/wiki/hershey%7s_kisses and candyprofessor.com The Hershey Kisses were first introduced in It is believed that the candy was named Kisses because of the sound and motion made by machine while depositing the chocolate. At first, the Hershey Kisses were wrapped by hand, but in 191, a machine was made so the Kisses would be wrapped automatically. This is also when the plume was added. In 194, Milton S. Hershey received a registered trademark for the plume. During 194, production of Hershey Kisses was briefly interrupted due to the rationing of aluminum foil. Instead, the machines were used to create chocolate paste for the soldiers in World War II. In 1976, the Kiss received a registered trademark for the foil wrapper. Kisses are one of the most popular brands of candies in the US. In 1989, the chocolate drops were the 5th most popular chocolate brand in the United States, spawning sales that topped $400 million. More than 60 million Hershey's Kisses chocolates are produced each day at the company's two factories. Today's Kisses brand chocolates use Hershey's original milk chocolate formula. Hershey s today is one of the major candy companies in the world, boasting annual sales in excess of $4 billion dollars. But around 1900, Hershey s was one among many contenders for America s top chocolate maker. The big business in chocolate at that time was not so much direct retail products, but selling various coatings and chocolate ingredients to candy makers large and small. Rivals like Stollwerck Brothers of New York and Chicago, H. O. Wilbur and Sons of Philadelphia, and Rockwood and Co. of Brooklyn were promoting their own chocolate goods, each promising purity, quality and taste unrivaled. Finished candy goods were, for many of these companies, a side line to the real action in wholesale cocoa and chocolate. Rival chocolate manufacturer H.O. Wilbur and Son had been selling a bite-sized foilwrapped conical chocolate drop called the Wilbur Bud since You wouldn t know it today, but back in the 1900s, Wilbur set the standard for those little foil-wrapped chocolates. The candy journal International Confectioner waxed rhapsodic over the beauty and hygiene of Wilbur s candy in 1914: Each piece is wrapped separately; they are packed like jewels. A large box of Wilburbuds can lie open several days before it is all eaten. Our little Wilburbuds can t go stale each is wrapped in foil. It was Hershey that was the copycat. And Hershey wasn t the only one. H.O. Wilbur even went to court in 1909 to try to stop the imitators. One of these might have been Rockwood s Chocolate Dainties, which were sold four for one cent. In their little foil wrapper, they would have been indistinguishable from Wilbur s Buds or Hershey s Kisses or any other similar chocolate. 3

4 Unwrapped, the Wilbur Bud was quite distinctive; the bottom of the candy was molded into a flower shape and the letters W-I-L-B-U-R embossed in each petal. In contrast, the Hershey s Kiss then as now isn t much to look at. It is just a plain cone, its bottom flat and unadorned. While this perhaps was less lovely to behold, it did mean the Kiss could be manufactured by dropping the chocolate on a flat belt, rather than needing special molds. This would eventually matter quite a lot, but in 1907 the Kiss s plain-jane looks would have been a distinct disadvantage. The decisive moment for the Hershey s Kiss was 191, when new manufacturing equipment allowed the foil wrapping to be automated, and also allowed for the inclusion of the plume that extends from the top of every Chocolate Kiss. By spring of 19 Hershey s was taking out full page ads blaring Insist upon having the GENUINE Sweet Milk Chocolate Hershey s KISSES. Be Sure They Contain the Identification Tag HERSHEY S. The plume was trademarked in 194, meaning that no other conical foil wrapped chocolate could use the same technique to stand out. Wilbur, and many other small American chocolate concerns, eventually fell behind Hershey in the race for market share. Milton Hershey was a generous philanthropist as well as a brilliant business man, and the success of the company in dominating the American chocolate scene is a fascinating story of doing well by doing good. Today, Hershey s Kisses are a multi-million dollar share of the American candy market. And Wilbur s Chocolate Buds? You can still buy them by mail-order, or out of a little Wilbur Chocolates storefront in Lititz, Pennsylvania. 4

5 Part The Hershey Kiss Model Imagine taking a Hershey Kiss, cutting it half, and placing this half-kiss on a coordinate grid such that the flat, wide base of the Kiss aligns with the vertical axis. Then imagine tracing the outline of the Kiss to create the shape of the outline of this Kiss with positive vertical axis values. When the area below these data points (when a curve is created to fit the data points) is revolved around the horizontal axis, a Hershey s Kiss is formed as shown below (where the grid is ¼ cm square): Drawing not to scale grid is ¼ cm square 1. Using the regression features available on a graphing device, determine a function formula for the graph that models the outline of the Hershey Kiss as shown in the image. Write a clear, mathematical argument explaining why you chose the particular function chosen to model this Hershey Kiss situation. 5

6 . Using your model in #1, set up, but do not evaluate, an integral that could be used to find the volume of your Hershey Kiss. 3. Using technology, evaluate the integral you created in #. Be sure to label your result with appropriate units. 4. Report the volume of the Hershey s Kiss in cubic centimeters of chocolate if you haven t already. 6

7 Part 3 Wrapping the Hershey s Kiss 1. Determine the minimum amount of foil needed to wrap the Hershey s Kiss. We say minimum amount here since there is clearly some overlap when wrapping chocolate. However, your task is to find the surface area (including the base) so that we have a baseline figure for the amount of foil needed to wrap each Kiss.. Suppose that the foil wrapper begins as a perfectly square piece of silver foil. What are the dimensions of this piece of foil? Express your answer rounded to the hundredths place. 7

8 . Hershey s reports that more than 60 million Hershey s Kisses are produced each day at the company s two factories. How many times (including fractional number of times) could the Washington Monument be wrapped in the foil needed to wrap one-year worth of Hershey s Kisses? 8

9 Part 4 World s Largest Hershey s Kiss According to the largest individual chocolate was a chocolate Hershey s Kiss weighing 13,85.71 kg (30,540 lbs.). The chocolate was made to celebrate Hershey's Kisses 100th anniversary and was displayed at Chocolate World, Hershey, Pennsylvania, USA, on 7 July 007. According to the density of chocolate is 135 kg/m 3 plus (+) or minus (-) 1%. Chocolate does not absorb or dissolve any substantial amount of water. Therefore, displacement method can be used to determine its volume, and with the mass available, its density can be calculated. 1. Based on your work in Part, how many regular Hershey s Kisses would it take to make the world s largest Kiss? Clearly show all the work needed to answer this question.. Suppose the function f() t represents the number of pounds of chocolate contained in the world s largest Kiss t hours after the beginning of construction. Explain what f (6) 150 means in this context. What units would be used to explain f (6) 150? 3. Continuing from # above, suppose f (6) 500 and f (6) 150. Estimate f (4.5) and explain what your result means. 9

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