2010 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 2w
Perimeter of a rectangle
A
r2
Area of a circle
y2 y1
x2 x1
Slope
Circumference of a circle
1
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
s
A
C 2 r
m
b
1
2
f ( x) dx
a
Arc length
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.
V
r 2h
V
Area of Base height
Volume of cylinder
Lateral SA = 2
Volume
r h
Lateral surface area of cylinder
b 2 4ac
2a
Quadratic Formula
x
b
4 3
r
3
Volume of a sphere
V
tan
sin
cos
References: www.usbr.gov
www.enchantedlearning.com
www.associatedcontent.com
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TEAM PROJECT Level II
2010 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.
Flying Under the Gateway Arch in St. Louis, Missouri – A Hypothetical Situation
The St. Louis Visitors and Convention Bureau has contacted you about a new idea that is being pitched to the
Bureau next month. A local airplane pilot, Charles, has requested permission to conduct tourist flights through the
Gateway Arch. According to www.enchantedlearning.com, “the St. Louis Gateway Arch is an elegant monument
to westward expansion in the USA. Located on the banks of the Mississippi River in St. Louis, Missouri, the 630foot tall stainless steel arch rises above the city skyline.” Charles would like to take tourists on a thrilling flight
passing through the Gateway Arch. The Bureau, considering the safety of such a stunt, wants to be sure that the
plane Charles will be using will be able to safely pass through the arch at the altitude proposed by Charles. Charles
has indicated that he will fly through the arch at an altitude of 600 feet, safely passing through the arch. Can this
be done? To answer this question, complete the following activities.
Part I – Graphical Investigation
1. Using the image of the Gateway Arch below to develop a function that models the shape of the arch. The input
variable, h, is the horizontal distance from the left edge of the arch and the output variable, v, is the vertical
height of the arch. For this part of the project, assume that the arch follows a parabolic path and express your
function in standard form ( v f (h) a h 2 b h c ).
v
h
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2. Sketch a graph of your function. Be as accurate and neat as you can be.
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3. Using the graph you just created and the fact that the airplane that Charles will fly through the arch has a
wingspan of 36 feet, determine if the flight can be performed safely. Remember that Charles would like to fly
through the arch at an altitude of 600 feet. Write several sentences describing your ideas. Show your ideas
visually using the graph you just created.
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Part II – Algebraic Investigation
1. Since pilot Charles has indicated that he plans to fly the plane through the arch at an altitude of 600 feet, write
an equation that would determine the values of h on the parabola that would produce a v-value of 600.
2. Solve the equation found you created above. Describe what your result means in the context of this problem
situation.
3. Use the results from number 2 above to determine the horizontal distance across the Gateway Arch at the 600
foot height.
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4. Charles’s plane, a Cessna 180, has a wingspan of 36 feet. Do you think that if he is to fly his plane through the
arch at an altitude of 600 feet that there will be enough clearance for the plane to make it through? Explain.
5. Carefully analyze the relationship between the solutions to your equation from number 2 above and the vertex
of the parabola (i.e. the arch). What relationship to you notice?
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Part III – Analyzing Rates of Change
1. Consider the function v
a h2
b h c found in Part I of this project. On the blank grids below, sketch
dv
the graph of the function v f (h) and its derivative
f (h) . Clearly label points and/or areas that you think
dh
are most significant in this problem situation and write a sentence explaining what these points and/or areas
represent and why you claim them to be significant. Specially, explain why Charles might find this analysis to
be significant.
v
dv
dh
f ( h)
f ( h)
f ( h)
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2. Find
dv
dh
f (250) and explain what information it provides in the context of this situation. Then find
dv
f (500) and explain what information it provides in the context of the situation. Specifically, explain how
dh
this information might be best interpreted by the pilot, Charles.
dv
f (h) , provides in the context
dh
of the situation. Like in number 2 where you described a particular rate of change computation, you will now
explain what the rate of change function means in the context of this situation. Specifically address Charles and
explain what particular information the derivative function provides him in the context of flying the plane
through the arch.
3. Write several sentences explaining what information the derivative function,
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4. What is the maximum altitude that Charles could fly and still make it safely through the arch? Show all the
work that supports your answer. It is advisable to include graphs to help support your reasoning and your final
result. Also discuss any real world considerations that might be made as you think about the mathematics
involved in this problem situation.
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Part IV – Perspective Regarding the Size of the Arch
In this part, you will approximate the volume of the arch. That is, imagine that you could fill the arch with water.
How much water would it take to fill the arch? Below are several facts that may prove to be useful for this part of
the project.
Arch consists of 142 equilateral triangular sections.
The equilateral triangles at the base have sides of length 54 feet.
The equilateral triangles at the top of the arch have sides of length 17 feet.
Completed in October 1965, he cost to build the Arch was $13 million.
The usual sway of the Arch is 1/2" and the maximum sway in a 150 mph wind is 9" each way.
Nine hundred tons of stainless steel was used to build the Arch and it weighs 17,246 tons.
1. Set up a definite integral that will give the total length of the arch (as if we could straighten it out and measure
the total length of the arc formed by the arch).
v
h
2. Evaluate the integral you created in #1 above. You may use technology.
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3. Set up an integral that will determine the total volume of the St. Louis Arch. Note that the known cross-sections
are equilateral triangles. You may assume that that lengths of the sides of these equilateral triangles changes
linearly from the bottom of the arch to the top of the arch. Relevant measurements are given in the fact box at
the beginning of Part IV.
4. Evaluate the integral you created in #3 above. You may use technology.
5. According to www.usbr.gov, the Hoover Dam was constructed using 3.25
million cubic yards of concrete. How many times could you fill the St. Louis
Arch using the amount of concrete used to build the Hoover Dam? You may
take an average of the larger volume and lesser volume found previously.
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Part V – Actually a Catenary
Because of the physics involved in the St. Louis arch, we can model its shape more accurately as a catenary rather
than as a parabola. The simplest way to think about a catenary is to imagine holding the ends of a string and
allowing the string to hang.
This catenary can be modeled using exponential growth (the right side of the catenary) and exponential decay (the
left side of the catenary).
This type of function is used often enough that it is given its own name…it is a hyperbolic cosine (cosh) function
and is defined as follows:
cosh(ax)
eax
e
2
ax
The St. Louis Arch is actually in inverted catenary and is best modeled by the function
v
f (h)
68.8cosh(0.01h 3.15) 700 .
dv
dv
dv
f (h) . Then find
f (250) and
f (500) .
dh
dh
dh
Compare these derivative computations with those found in Part III. What does this tell you about the difference
between the parabolic model created earlier and this hyperbolic cosine (catenary) model?
Work to develop a formula for the derivative function,
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Part V Continued…
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GRADING RUBRIC FOR St. Louis Arch Project
High School Mathematics Contest - 2010
WORK/RESULTS
Correct computations throughout
o Part I
 Quadratic Function
 Accurate Graph
o Part II
 Set up equation
 Solve equation
 Determine horizontal distance
 Correct vertex
o Part III
 Correct graphs
 Correct interpretation of derivative function
 Correct interpretation of specific derivative function values
 Correct maximum height computation
o Part IV
 Correct integral set up for total length of arch
 Correct evaluation of the integral for the length of arch
 Correct set up of volume integral
 Correct evaluation of integral
 Correct comparison to Hoover Dam
o Part V
 Correct derivative of the hyperbolic cosine function
 Correct evaluation of derivative at specified values of h
 Appropriate comparison to parabolic model
Accurately written interpretations of mathematical ideas
o Part I – explain how graph is used to address problem situation
o Part II – explain relationship between vertex and vertical intercepts
o Part III – accurate description of rate of change value and function/graph
o Part V – accurate comparison of catenary and parabolic models
TOTAL……….
____/5
____/10
____/10
____/15
____/5
____/15
____/60
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