2016 Excellence in Mathematics Contest
Team Project Level I
(Precalculus and above)
School Name:
Group Members:
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ANSWERS
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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  2 r
Circumference of a circle
C  2l  2w
Perimeter of a rectangle
A   r2
Area of a circle
b
1
A  bh
2
Area of a triangle
V   r 2 dx
5280 feet = 1 mile
V   2rhdx
a
Volume
b
a 2  b2  c 2
a
Pythagorean Theorem
2
 dy 
s   1    dx
 dx 
a
b
Volume
2.54 centimeters = 1 inch
h  4.9t 2  v0t  h0
h  16t 2  v0t  h0
Arc Length
1 kilogram = 2.2 pounds
1 meter = 39.3701 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.
V   r 2h
Volume of cylinder
Lateral SA = 2  r  h
Lateral surface area of cylinder
V   Area of Base   height
Volume
 b  b 2  4ac
x
2a
Quadratic Formula
4
V   r3
3
Volume of a sphere
b
V    ( R 2  r 2 )dx
a
Volume
2
TEAM PROJECT Level I
2016 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. We encourage the use of a graphing calculator.
Part 1: Background
Sources: en.wikipedia.org/wiki/Hershey%27s_Kisses and candyprofessor.com
The Hershey Kisses were first introduced in 1907. 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 1921, a machine was made so the Kisses would be wrapped automatically. This
is also when the plume was added. In 1924, Milton S. Hershey received a
registered trademark for the plume. During 1942, 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 1894. 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.
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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
1921, 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
1922 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 1924, 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.
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Part 2 – 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.
I will show a desmos.com, wolframalpha.com solution. Note that students will likely be limited to their TI-84 or
similar.
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2. 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 #2. Be sure to label your result with appropriate units.
The volume, according to this polynomial model is approximately 3.14962 cubic centimeters or 3.14962 mL
4. Report the volume of the Hershey’s Kiss in cubic centimeters of chocolate if you haven’t already.
The volume, according to this polynomial model is approximately 3.14962 cubic centimeters or 3.14962 mL
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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.
Lateral surface area using my polynomial model is about 8.84829 sq. cm.:
The bottom is just a circle of radius 1 cm:
A   1  3.14159 sq. cm.
2
Total area then is about 8.84829 + 3.14159 = 11.9898 sq. cm.
2. 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.
Each piece of foil is approximately 11.9898 cm by 11.9898 cm or about 3.46 cm by 3.46 cm square.
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2. 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?
Surface area of trapezoidal sides:
1

S . A.  4   500   55  34.5   89,500 sq. ft.
2

Surface area of pyramid sides at top:
1

S . A.  4   34.5  50   3, 450 sq. ft.
2

Total surface area of monument is 89,500 sq. ft. + 3,450 sq. ft. =
92,950 sq. ft.
Surface area of a foil Kiss wrapper is about 12 sq. cm. or about
0.01292 sq. ft.
How many 0.01292 sq. ft. Kiss wrappers are in 92,950 sq. ft.?
92950
 7,194, 272.4458
0.01292
Over 7,194,272 Kiss wrappers are needed to wrap the monument.
At 60 million a day, there are 60,000,000 times 365 or 21,900,000,000 Kisses made each year and thus
21,900,000,000 wrappers created each year.
With 21,900,000,000 wrappers, the monument could be wrapped
21,900, 000, 000
 3044 times!
7,194, 272
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Part 4 – World’s Largest Hershey’s Kiss
According to www.guinnessorldrecords.com, “the largest individual
chocolate was a chocolate Hershey’s Kiss weighing 13,852.71 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 2007.”
According to www.quora.com, “the density of chocolate is 1325
kg/m3 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 2, 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.
The volume of a Kiss is 3.14962 cubic centimeters or
3.14962x10 –6 cubic meters making its mass
kg
1325 3  3.14962 106 m3 or about 0.004173 kg.
m
How many Kisses are needed to make the 13,852.71 kg giant Kiss? About
13852.71
 3,319, 604.601 Kisses!
0.004173
2. 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 ?
f (6)  150 means that 6 hours after the start of construction, the amount of chocolate contained in the giant
Kiss is increasing at a rate of 150 pounds of chocolate per hour. The units are “pounds of chocolate per hour.”
3. Continuing from #2 above, suppose f (6)  500 and f (6)  150 . Estimate f (4.5) and explain what your result
means.
f (4.5)  f (6)  1.5 f (6)
 500  1.5 150
 275
If the amount of chocolate was increasing at a constant rate, then after 4.5 hours (1.5 hours before the Kiss has
grown to 500 pounds), the Kiss would have accumulated 275 pounds of chocolate. Note to judges…a top
scoring team would include the idea of “constant rate” in their explanation.
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