2015 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  2 r
Circumference of a circle
a 2  b2  c 2
Pythagorean Theorem
A   r2
Area of a circle
2
1
A  bh
2
Area of a triangle
 dr 
s   r 
 d
 d 
a
Arc Length
5280 feet = 1 mile
3 feet = 1 yard
2.54 centimeters = 1 inch
h  4.9t 2  v0t  h0
b
2
2
 dy 
s   1    dx
 dx 
a
Arc Length
b
C  2l  2w
Perimeter of a rectangle
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 
2
TEAM PROJECT Level I
2015 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
In June 2013, a 1998 Corvette* raced down the quarter-mile
drag strip at the Route 66 Raceway in Joliet, Illinois. The
car was at the starting line waiting for the signal light to turn
green at which time the driver reacted, stomping on the car’s
accelerator, and racing down the track. At the end of the
race, a printout of data was provided to the driver (see
below). Focus on the data for Car #585.
Legend:
R/T – Reaction Time
E.T. – Elapsed Time
*Murawska, J. and Nabb, K. (2015). Mathematics Teacher Vol. 109, No. 2, September 2015. National Council of Teachers of Mathematics
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Part 2: 60 MPH
For sports cars, a common measure of performance is the number of seconds
it takes the car to accelerate from 0 to 60 mph. The driver of this Corvette
would like to know, according to these data, how many seconds it took him
to reach a speed of 60 mph. Your task is to determine this time and support
your claim mathematically. Include an explanation in words and a graph, if
necessary. Note that you should focus on the data for Car #585 and that you
should disucss what it means, in terms of your data, for the driver to have a
reaction time of 0.734 seconds. Furthermore, note that it is conventional for
drag strip data to provide the time at 60 feet, 330 feet, 1/8 of a mile, 1000
feet, and at the finish line (1/4 mile). The speed recorded at the finish line
was 116.14 mph.
This is a very open ended task with many possible solution strategies. You
will have to judge each team’s response for mathematical accuracy, effective
modeling approach, communication, and the ability to make a convincing argument.
A possible solution is shown below.
Fit a polynomial curve to the data and interpret this curve as the Corvette’s position. Use differentiation to find
instantaneous velocity and algebraically or graphically find the moment in time when the velocity is 88 feet per
second or 60 miles per hour.
Using the above model ( s  6t 2  38t  38 ), the derivative (velocity) function is v  12t  38 feet per second.
Therefore, the velocity is 88 feet per second (60 mph) when t  4.2 seconds.
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Part 2: Continued…
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Part 3: Maximum Speed
The image shows the speedometer for a Corvette. The
maximum speed shown by the speedometer is 200 mph.
Your task is to determine the amount of additional time
needed for the drag racing Corvette from Part 2 to reach
a speed of 200 mph. How long (distance) would the
race need to be for this to occur? Clearly state any
assumptions you make in responding to this issue.
Again, answers will vary based on the models and
strategies used in Part 2.
Using the previous model and assuming that the Corvette continues to accelerate according to this model, it would
take about 13.5 seconds to reach a speed of 200 mph. This would mean that the car would have had to travel an
additional 1.241 seconds after the end of the quarter-mile track.
v  12t  38
200  12t  38
12t  162
t  13.5
If the car traveled for 13.5 seconds, the model shows that the car would have traveled about 1566.4 feet or almost
three-tenths of a mile.
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Part 4: Constant Acceleration?
We will now make a new assumption and create some new data. Rather than the situation presented earlier, let’s
assume that the Corvette completed the quarter-mile drag race in such a way that its acceleration was constant.
1. At what constant acceleration would allow the Corvette to complete the quarter-mile race in 11.514 seconds
after a 0.682 second reaction time? That is, the total elapsed time is the sum of these two times. Futhermore,
assume you know that the time to reach the 1/8 mile mark was 7.522 seconds after the reaction time.
Students may recognize that they can create the model starting with the actual movement of the car after the
reaction time has elapsed. Therefore, position is s  0 when t  0 .
aK
v  Kt  C
1
s  Kt 2  Ct  D
2
1
2
0  K  0  C 0  D
2
0D
1
2
1320  K 11.514   C 11.514 
2
1
2
660  K  7.522   C  7.522 
2
1
So K  13.48 feet per second per second and s  13.48  t 2  37.06  t  6.74  t 2  37.06  t .
2
2. Give the constant acceleration determined in #1, how long would it take the Corvette to reach a speed of 60
mph?
v  13.48t  37.06
88  13.48t  37.06
t  3.78 seconds
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Part 5: Dragster vs. Jet Car
Suppose a traditional dragster and a jet dragster raced such that they
produced the acceleration graphs shown. The vertical dashed line
represents the final time at the end of the race.
1. Compare the velocities of the two cars at the end of the race. Clearly
explain your thinking.
Since the areas under the two accleration graphs appear to be equal, a
claim can be made that both cars have the same velocity at the end of
the race.
2. Assuming that the finish line is a quarter mile from the starting line, which car won the race? Clearly explain
your thinking.
Using the acceleration graphs given, students might sketch
velocity and/or position graphs. Based on the velocity graphs,
an arugment can be made that the dragster would produce a
velocity graph that is increasing and concave down. The jet car
would produce a velocity graph that is increasing but concave
up. With such a short race, the argument would include the
idea that the area under the curve for the dragster is greater
than that for the jet car so the dragster might get to the finish
line first.
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Part 5: continues…
3. In #2 above, you made a claim and an argument about either the jet car or the dragster winning the race. Now,
imagine a situation where the other car wins the race. Sketch acceleration graphs that would support the case
that the other car wins the race. That is, how might you change the acceleration graphs given in #1 (of this Part
5) so that the other car wins the race? In the space below, neatly show the graphs and write an explanation
detailing why your new graphs support the claim that the other car now wins the race.
Answers will vary. Check the reasonableness of the response and judge the effectiveness of the communication
supporting the response.
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