MTH 229 - Calculus with Analytic Geom I
Project 1
Project Instructions:
1. Mathematics:
Application of appropriate calculus skills and mathematics.
Calculus must be applied to each of the problems.
2. Representation:
Thorough Analytical and graphical explanations of problem.
3. Construction:
Constructions must be done with precise measurements.
4. Neatness:
All work must be legible and easy to understand.
Plagiarism is not allowed in this project work. You are not allowed to consult anyone or use any material except your textbook,
group member, or me. The report must contain all software and
other sources used.
Final Submission:
Final report of project work must be TYPED using MS-WORD, LATEX or any format and submitted online on GEAR. Instructions on how to
submit on GEAR is posted on blackboard under the tab ”Project”
1
Project 1:1
Pizza Hut has just introduced a larger personal size pizza. The Pizza Hut
Company needs a design for a box to package this new pizza. The box is
to be made from a piece of cardboard of size 10 inches by 18 inches. The
company wants to design a box with the largest volume. Six squares are to
be cut from the cardboard as shown in Figure 1. By using the application of
calculus, find the dimension of the box with largest volume. The company
will also like you to construct a prototype of the box with such dimensions
using a piece of cardboard
Figure 1:
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Project 1:2
A norman window has the outline of a semicircle on top of a rectangle
as shown in Figure 2. Suppose there is 9 + π feet of wood trim available
for all 4 sides of the rectangle and the semicircle, find the dimensions of the
rectangle and semicircle that will maximize the area of the window.
Figure 2:
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Project 1:3
Find the maximum area of a rectangle circumscribed around a rectangle
of sides x and y. (Hint: Express the area in terms of the angle θ.
Figure 3:
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Project 1:4
The corn yield on a certain farm is
Y = −0.118x2 + 8.5x + 12.9 bushels per acre
where x is the number of corn plants per acre (in thousands). Assume that
corn seed costs $1.25 (per thousand seeds) and that corn can be sold for
$1.50/bushel. Let P (x) be the profit (revenue minus the cost of seeds) at
planting level x.
a). Compute P (x0 ) for the value x0 that maximizes yiels Y .
b). Find the maximum value of P (x).
c). Does maximum yield lead to maximum profit?
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MTH 229 - Calculus with Analytic Geom I
Project 2
Project Instructions:
1. Mathematics:
Application of appropriate calculus skills and mathematics.
Calculus must be applied to each of the problems.
2. Representation:
Thorough Analytical and graphical explanations of problem.
3. Construction:
Constructions must be done with precise measurements.
4. Neatness:
All work must be legible and easy to understand.
Plagiarism is not allowed in this project work. You are not allowed to consult anyone or use any material except your textbook,
group member, or me. The report must contain all software and
other sources used.
Final Submission:
Final report of project work must be TYPED using MS-WORD, LATEX or any format and submitted online on GEAR. Instructions on how to
submit on GEAR is posted on blackboard under the tab ”Project”
1
Project 2:1
Ornithologists have determined that some species of birds tend to avoid
flights over large bodies of water during daylight hours. It is believed that
more energy is required to fly over water than over land because air generally rises over land and falls over water during the day/ A bird with these
tendencies is released from an island that is 5km from the nearest point B
on a straight shoreline, flies to a point C on the shoreline, and then flies
along the shoreline to its nesting area D. Assume that the bird instinctively
chooses a path that will minimize its energy expenditure. Points B and D
are 13 km apart.
a). In general, f it takes 1.4times as much energy to fly over water as it
does over land, to what point C should the bird fly in order to minimize the
total energy expenditure in returning to its nesting area?
b). Let W and L denotes the energy (in joules) per kilometer flown over
water and land, respectively. Determine the ratio W/L corresponding to the
minimum expenditure energy.
c). What should W/L be in order for the bird to fly
i). directly to its nesting area D?
ii). to B and then along the shoreline to D?
d). If the ornithologists observe that the birds of a certain species reach
the shore at a point 4 km from B, how many times more energy does it take
a bird to fly over water than over land?
Figure 1:
2
Project 2:2
A company plans to introduce a new right circular cylindrical open-top cup
for their soft drink. The cup is to hold 266 cubic centimeter of liquid when
filled to the top. The company wants your team to design a cup that can be
constructed with the least amount of material so that they can cut back on
production cost. What are the dimensions of the cup? The company will
also like you to construct a prototype of the cup with such dimensions using
a piece of cardboard.Construct the cup using a piece of cardboard.
3
Project 2:3
Your group is assigned to design a rectangular industrial warehouse consisting of three separate spaces of equal size as shown in Figure 2. The wall
materials cost $500 per meter and a company allocates 2, 400, 000 for this
project.
a). Which dimension maximize the area of the warehouse?
b). What is the area of each compartment in this case?
Figure 2:
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Project 2:4
In 1535 the mathematician Antonio Fior challenged his rival Nicolo
Tartaglia to solve this problem:
A tree stands 12 braccia high; it is broken into two parts at such a point
that the height of the part left standing is the cube root of the length of the
part cut away.
a). What is the height of the part left standing?
b). Show that this is equivalent to solving x3 + x = 12.
c). Find the exact height x of the part left standing to nine decimal
places. (state the method used). Show that your answer is equivalent to the
exact solution given by
p
p
√
√
3
3
2919 + 54 −
2919 − 54
√
x=
.
3
9
d). Find the exact height x of the part cut away to nine decimal places.
(state the method used).
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MTH 229 - Calculus with Analytic Geom I
Project 3
Project Instructions:
1. Mathematics:
Application of appropriate calculus skills and mathematics.
Calculus must be applied to each of the problems.
2. Representation:
Thorough Analytical and graphical explanations of problem.
3. Construction:
Constructions must be done with precise measurements.
4. Neatness:
All work must be legible and easy to understand.
Plagiarism is not allowed in this project work. You are not allowed to consult anyone or use any material except your textbook,
group member, or me. The report must contain all software and
other sources used.
Final Submission:
Final report of project work must be TYPED using MS-WORD, LATEX or any format and submitted online on GEAR. Instructions on how to
submit on GEAR is posted on blackboard under the tab ”Project”
1
Project 3:1
Your group’s job, as a consultant to a candy maker, is to design a jelly
bean boxes each having a volume of 1200 inch3 . Each of these jelly bean
boxes is to be an open-topped rectangular box with a square base having
edge length x. In addition, the box is to have a square lid with a 2-inch rim.
Thus the box-with-lid actually consists of two-opened boxes-the candy box
itself with height y-inch, and the lid with height 2 − in (we assume the lid
fits very snugly). If it costs $9 per square inch each to construct the opentopped rectangular box and the square lid. The candy maker’s problem is
to determine the dimensions x and y that will minimize the total cost of
production.
Figure 1:
The company will also like you to construct a prototype of the box with
such dimensions using a piece of cardboard.
2
Project 3:2
Figure 2:
3
Project 3:3
A box with a square base and open top (shown below) must have a
volume of 32, 000 cm3 . Find the dimensions of the box that minimize the
amount of material used (i.e. surface area).
Figure 3:
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Project 3:4
A box of volume 8m3 with a square top and bottom is constructed out
of two types of metal. The metal for the top and bottom costs $50/m2 and
the metal for the sides costs $30/m2 . Find the dimensions of the box that
minimize total cost.
5
MTH 229 - Calculus with Analytic Geom I
Project 4
Project Instructions:
1. Mathematics:
Application of appropriate calculus skills and mathematics.
Calculus must be applied to each of the problems.
2. Representation:
Thorough Analytical and graphical explanations of problem.
3. Construction:
Constructions must be done with precise measurements.
4. Neatness:
All work must be legible and easy to understand.
Plagiarism is not allowed in this project work. You are not allowed to consult anyone or use any material except your textbook,
group member, or me. The report must contain all software and
other sources used.
Final Submission:
Final report of project work must be TYPED using MS-WORD, LATEX or any format and submitted online on GEAR. Instructions on how to
submit on GEAR is posted on blackboard under the tab ”Project”
1
Project 4:1
2
Project 4:2
The rectangular plot (Figure below) has size 100m × 200m. Pipe is to
be laid from A to a point P on side BC and from there to C. The cost of
laying pipe along the side of the plot is $45 per meter and the cost through
the plot is $80/m (since it is underground).
a). Let f (x) be the total cost, where x is the distance from P to B.
Determine f (x), but note that f is discontinuous at x = 0 (when x = 0, the
cost of the entire pipe is $45 per feet)
b). What is the most economical way to lay the pipe? What if the cost
along the sides is $65 per meter?
3
Project 4:3
A swimmer launches his boat from point A on a bank of a straight river,
3km wide, and wants to reach point B, 8 km downstream on the opposite
bank, as quickly as possible. He could proceed in any of three ways:
1. Row his boat directly across the river to point C and then run to
point B
2. Row directly to point B
3. Row to some point D between C and B and then run to point B.
If he can row 6km/h and run 8 km/h, where should he land to reach
point B as soon as possible?
Figure 1:
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Project 4:4
A truck gets 10 miles per gallon (mpg) of diesel fuel traveling along an
interstate highway at 50 mph. This mileage decreases by 0.15 mpg for each
mile per hour increase above 50 mph.
a). If the truck driver is paid $30/h and diesel fuel costs P = $3/gal,
which spead v between 50 and 70 mph will minimize the cost of a trip along
the highway? Notice that the actual cost depends on the length of the trip,
but the optimal speed does not.
b). Plot cost as a function of v (choose the length arbitrarily) and verify
your answer to part (a).
c). Do you expect the optimal speed v to increase or decrease if fuel
costs go down to P = $2/gal?
5
MTH 229 - Calculus with Analytic Geom I
Project 5
Project Instructions:
1. Mathematics:
Application of appropriate calculus skills and mathematics.
Calculus must be applied to each of the problems.
2. Representation:
Thorough Analytical and graphical explanations of problem.
3. Construction:
Constructions must be done with precise measurements.
4. Neatness:
All work must be legible and easy to understand.
Plagiarism is not allowed in this project work. You are not allowed to consult anyone or use any material except your textbook,
group member, or me. The report must contain all software and
other sources used.
Final Submission:
Final report of project work must be TYPED using MS-WORD, LATEX or any format and submitted online on GEAR. Instructions on how to
submit on GEAR is posted on blackboard under the tab ”Project”
1
Project 5:1
Find a point C on the circle of radius 1 centered at the origin that makes
the triangle ABC of largest possible area, given that A and B are the points
(0, 2) and (3, 0)
2
Project 5:2
Find the maximum area of the rectangle inscribed in the region bounded
by the graph of y = 4−x
2+x and the axes. (see Figure below).
3
Project 5:3
A four-wheel-drive vehicle is transporting an injured hiker to the hospital
from a point that is 30km from the nearest point on a straight road. The
hospital is 50 km down the road from the nearest point. If the vehicle can
drive at 30kph over the terrain and at 120kph on the road, how far down
the road should the vehicle aim to reach the road to minimize the time it
takes to reach the hospital?
4
Project 5:4
What is the maximum area of a rectangle inscribed in a right triangle
with legs of length 3 and 4? The sides of the rectangle are parallel to the
legs of the triangle.
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