Student Research Projects in Calculus - 2001
Here are the choices for you for your year-end Calculus projects. You may read them all; I suggest you do.
However, each was chosen to be no easier or harder than the others. A pair of students working together on
these projects should complete the work in 1 week.
Tent Makers
Imagine you are to manufacture a tent. You have two choices from which to select a basic design. One is
called the “spherical cap”. It is a spherical dome with the bottom sliced away by a plane floor. Two is
called the “pup tent” that consists of a right prism whose cross section is an
equilateral triangle. Each tent is to have a single door that shuts without overlap.
Assume we want the volume to be 2.2 m3, and we want to accommodate 3
sleeping people. The floor of the tent is cheaper material than the rest: assume that
the material making up the top is 1.40 times more expensive than the floor.
(a) What should be the dimensions of the tent to keep the cost to a minimum?
(b) What is the total are of material needed?
Now change the problem so that the floor of the tent is in fact 1.40 times more expensive that the rest.
(c) What should be the dimensions of the tent so that the cost of the materials used is a minimum?
(d) What is the total area of material used?
(e) How practical would each of the 4 tent designs be?
Finding YOUR area
In this project you are going to accurately trace the shape of a member of your group and
approximate the area of the picture of their whole body and his/her hand. Your main tasks
are to devise a method for approximating the areas and to show that you approximation is
very close to the actual area.
To begin with, start with a sheet of paper that is seven feet by three feet. Have a member of
your group lie on the paper and make an accurate trace of the outline of that person. You
will have to hand in this paper with your project.
Devise a method to approximate the area of the region inside the tracing. You must explain
your method in detail and why it works.
Using the method devised above, find the area of the two regions that differs from the
actual area by less than 1%. This is an important part of the project. You should spend
some time considering how accurate an answer you require. Reading about “relative
errors” might be of help. You will not know the actual value for the area, so you must
devise a way to show that your approximation is within the desired degree of accuracy.
“Houdini’s Escape”
Harry Houdini was a famous escape artist. In this project we relive a trick of
his that challenged his mathematical prowess, as well as his skill and bravery.
It will challenge these qualities in you as well.
Houdini had his feet shackled to the top of a concrete block that was placed on
the bottom of a giant laboratory flask. The cross-sectional radius of the flask,
measured in feet, was given as a function of height z from the ground by the
formula r ( z ) 
10
, with the bottom of the flask at z  1 foot. The flask
z
was then filled with water at a steady rate of 22 cubic feet per minute. Houdini’s job was to escape the
shackles before the rising water in the flask drowned him. Now, Houdini knew it would take him exactly
10 minutes to escape the shackles before he would be drowned. For dramatic impact, he wanted to time his
escape so it would be completed precisely at the moment the water level reached the top of his head.
Houdini was exactly six feet tall. In the design of the apparatus, he was allowed to specify only on thing:
the height of the concrete block he stood on.
(a) You first task is to find out how high this block should be. Express the volume of the water as a
function of the height of the liquid above the ground level. What is the volume needed to reach the top
of his head? (Neglect Houdini’s volume and the volume of the block.) What is the height of the
block?
Let h(t) be the height of the water above the ground at any time t. In order to check the progress of his
escape moment by moment, Houdini derives the equation for the rate of change
dh
as a function of h
dt
itself. Derive this function. How fast is the water rising when it first starts to fill? How fast is it rising
when it reaches the top of his head? Express h(t).
(b) Houdini would like to be able to perform this trick with any flask. Help him by generalizing the
derivation. Consider a flask with cross sectional radius r(z) and a constant flow rate of
Find
dV
A.
dt
dh
as a function of time.
dt
“Quality Control”
I own a plant that manufactures disk drives.
My problem is to decide how many hours of labor to put into each drive.
The minimum labor time required to get a drive is 5 hours, but a 5-hour
drive will fail 10% of the time while in warranty. Any extra labor
time will reduce this failure probability at an exponential rate:
Specifically, each additional hour of labor cuts the failure rate by
40%. A failed drive is returned for repair; this costs us 2 hours of
additional labor. In addition, a repair incurs a cost of $200 for
shipping, handing, and damage to my companies reputation. My
labor cost is $7 per hour.
Your report is not limited to the following questions but should include these: 1) How many hours of labor
would minimize the cost per machine? 2) Suppose the hourly labor cost is w dollars per hour instead of $7
per hour; answer the same question. 3) The labor cost w is a parameter, and the optimum labor time
depends on w. Exactly what is the nature of this dependence? For what w should we go with a basic 5hour drive? For what w can we afford to increase the time put into a drive’s assembly?
Wine Barrel Capacity
A wine barrel is stored laying on its side and has a hole in the middle of its side called a “bung hole”. To
determine the volume of the wine in the barrel, a “bung rod” is inserted in the hole so that it hits the lower
seam at one end of the barrel. Determine a way to calibrate the rod so that is can be used to determine the
volume of wine in the barrel.
You should make the following assumptions:
1.
The barrel is a perfect cylinder.
2.
The distance from the bunghole to the lower corner seam is .
3.
The ratio of the height to the diameter of the barrel is t. This ratio is chosen so that for a given value of
, the volume of the barrel is maximal.
Your report is not limited to the following questions, but it should address these concerns. You should
show that the volume of the barrel is given V  2 t (4  t )
and give an approximate ideal value
for t. The famous astronomer Johannes Kepler was the first to explain the old cooper’s rule “make the
staves one and a half times the diameter.” (This is the t-value.) Provide dimension for the barrel and the
bung rod.
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2
3 / 2
Cockroach on an Elastic Tightrope
Before beginning the main parts of this project, answer the following
questions? Suppose you are standing on an elastic tightrope b meters
from the left end and c meters from the right end. Then suppose the
entire rope stretches uniformly, increasing its length by d meters. How
far are you from each end now? After answering this, you are ready to
begin.
Suppose a cockroach starts at one end of a 1000-meter tightrope and
runs to the other end at one meter per second. At the end of each
second, the entire rope stretches uniformly and instantaneously,
increasing its length by 1000 meters each time.
1.
Does the roach ever reach the other end?
2.
If so, how long will it take? Or prove that he can’t make it – ever!
Space Capsule Design
You are part of an engineering team designing the Apollo space capsule. The capsule is composed of two
parts:
1.
A cone with a height of 4 meters and a base radius of three meters;
2.
A re-entry shield in the shape of a parabola revolved about the axis of the cone, which is attached to
the cone along the edge of its base. The vertex of the parabola is to be a distance D below the base of
the cone at its axis.
Assume the capsule has a uniform density . Your project director has specified that the center of mass of
the capsule should be below the center of mass of the displaced water because he feels this will give the
capsule stability in high seas. He has given your team the task of finding values for D and  so that the
capsule will float with the vertex pointing up and with the waterline 2 meters below the top of the cone, in
order to keep the exit port
1
meter above the water.
3
Show your project director that this is impossible; i.e., there are no values of D and  that satisfy the
specifications.
You are locked in to three criteria:
I.
Archimede’s Principle: A body floats in a fluid at the level at which it displaces a weight of fluid
equal to the weight of the body.
II.
Consider the region bounded by the graph x=f(y), the y-axis, and the lines y=c and y=d. Derive a
formula for the center of mass of the solid formed by revolving this region about the y-axis.
Explain why the center of mass is on the y-axis so you just have to compute
III.
y.
Suppose that a body is made of two pieces, which are solids of revolution about the same axis.
Make a conjecture and prove how the center of mass of the composite body relates to the centers
of mass of the pieces.
Security System Project
You are designing a security system for the CIA. You must decide how to program a detector that will be
used to watch a 40-foot-long hallway with a door in the middle. The detector runs on a track and points a
laser beam of light straight ahead on the opposite wall. The beam reaches from floor to ceiling. Design a
security system that will detect an intruder.
You should make the following assumptions:
 Think of the hallway as a coordinate line with the middle of the door at the origin and the hallway
to be secured on the interval [-20,20]. You need to decide what x(t) is for t, in seconds, where x(t)
is the position of the beam at time t. For example, x(5) = -15 means the beam is pointing at the
part of the wall 15 ft to the left of the door 5 seconds after the detector starts its program.
 Assume that t ranges from 0 to 30 minutes.
 The door is 3 feet wide, and as long as the beam is hitting any part of the door it is under
surveillance.
 The beam must stay on an object for at least 0.1 seconds in order to detect it.
Your paper is not limited to the following questions but should include these concerns. Draw a graph of x
versus t for what you think is a good choice for x(t). You should mention any restrictions on x(t), if needed,
and include the longest time the door will not be under surveillance. Could an intruder get to the door by
walking down the hallway without being detected by your system? Explain how an intruder could do it and
how likely you think it is.