2012 Excellence in Mathematics Contest
Team Project Level I
(Precalculus and above)
School Name:
Group Members:
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
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  2w
Perimeter of a rectangle
A   r2
Area of a circle
y2  y1
x2  x1
Slope
1
A  bh
2
Area of a triangle
m
a 2  b2  c 2
Pythagorean Theorem
5280 feet = 1 mile
3 feet = 1 yard
16 ounces = 1 pound
2.54 centimeters = 1 inch
h  4.9t 2  v0t  h0
C  2 r
Circumference of a circle
h  16t 2  v0t  h0
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
V   Area of Base   height
Volume of cylinder
Lateral SA = 2  r  h
Lateral surface area of cylinder
Volume
 b  b 2  4ac
2a
Quadratic Formula
x
4
V   r3
3
Volume of a sphere
tan  
sin 
cos 
This team project is taken from The Mathematics Teacher, volume 105, Number 5, December 2011, National
Council of Teachers of Mathematics (nctm.org)
2
TEAM PROJECT Level I
2012 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.
During a visit with his sister’s family, Ron
Lancaster was shown an unusual bottle of
Coca-Cola® that consisted of a sphere
with a cap. Placing this bottle beside a can
of Pepsi cola revealed the contrast (see
photograph). Ron’s nephew Matt
challenged Ron to pick the container that
held the most liquid without touching
either or making any measurements. Ron
studied the bottle and can from a distance,
picked the one he thought had the greater
volume, and then found out he was
wrong. Ron then mailed his colleague
Doug Wilcock the spherical bottle with
this challenge: Set it beside a Pepsi can
and choose the container with the greater
volume. Not only did Doug pick the right
container, but he also devised the following set of questions related to the bottle and the can. Your task in this team
project is to respond to these questions as clearly and accurately as possible. Have fun!
Part 1 – Begin the Exploration
If you were given the same challenge as Ron Lancaster, which container would you choose as having the greater
volume? That is, without making any measurements or calculations, what does your gut instinct say? Explain.
3
Part 2 – With Measurements and Calculations
1. We can think of the Pepsi can as being a cylinder with a top in the shape of a frustum (A frustum is the portion
of a cone or pyramid that remains after its upper part has been cut off by a plane parallel to its base. See the
figure below). Use the measurements provided to determine the volume of the can.
4
Part 2 continued…
2. We now request that you use calculus to find the volume of
the Coca-Cola bottle. If your team does not have a member
with the necessary calculus background, then estimate the
volume of the bottle by assuming it is a perfect sphere.
a. A cross-section view of half the Coca-Cola bottle along
with its measurements is shown in the figure. Use calculus
to estimate the volume of a solid of revolution that models
the spherical bottle.
5
Part 2 continued…
b. The answer for part a. over-estimates the actual volume of
the contents of the Coca-Cola bottle. The reason: The base
is not flat but indented to provide more stability. A side
view of the indentation is shown in the figure. Measuring
indicates that AF  0.96 cm and AE  1.9 cm. What is the
volume of the indented section?
6
Part 2 continued…
c. If we take the calculated volume of the spherical bottle and subtract the volume of the indentation, we get
the approximate net volume of the bottle. Doing so, can we reach a conclusion about the relative sizes of the
spherical bottle and the cylindrical can? Explain.
7
Part 3 – Selling Soft Drinks
Suppose that Coca-Cola were to sell these special bottles in packages of six arranged as shown in the photograph
(two rows of three bottles).
1. If we define area efficiency as the ratio of the area of the bottles to the area of the container that will hold the
bottles (see the figure below), how efficient is the rectangular six-pack? Express your answer as a simple ratio.
8
Part 3 – continued…
2. A second way to consider efficiency is to consider the ratio of the volume of the liquid in the containers to the
volume of the packages (volumetric efficiency). The volume of soda in each bottle is 400 ml (400 cm3). The
bottles are 11.4 cm high. What is the volumetric efficiency of the rectangular six-pack?
9
Part 3 – continued…
3. Another way of packaging the six bottles is to arrange
them in the shape of a triangle (see photograph 3).
(a) Determine the area efficiency of the triangular sixpack (see fig. 5).
(b) Determine the volumetric efficiency of the triangular
six-pack.
10
Part 3 – continued…
4. A creative idea for packaging the six bottles is to stack them in pairs to form barbells (see photograph 4).
(a) What is the area efficiency of the barbell six-pack? In
answering this question, remember that there are six bottles,
not simply the three we see when we look at a plan of the
package, as shown in figure 6.
(b) What is the volumetric efficiency of the barbell six-pack?
11
Part 3 – continued…
5. Suppose that Coca-Cola decided to sell these unusual
bottles in a highly original pack of seven, arranged as
shown in photograph 5.
(a) What is the area efficiency of the heptahex-pack (see
fig. 7)?
.
(b) What is the volumetric efficiency of the heptahex-pack?
12
Part 3 – continued…
6. What type of package might you use for eight bottles?
13