Math125: Unit 1 Submission Assignment Answer Form

This article discusses how to solve problems related to sending a package and painting a bedroom using mathematical modeling and problem-solving techniques. It includes step-by-step calculations and formulas for finding dimensions, surface area, and volume.

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MATH125: Unit 1 Submission Assignment Answer Form
Mathematical Modeling and Problem Solving
ALL questions below regarding SENDING A PACKAGE and PAINTING A BEDROOM must be answered.
Show ALL step-by-step calculations, round all of your final answers correctly, and include the units of
measurement. Upload this modified Answer Form to the intellipath Unit 1 Submission lesson. Make sure
you submit your work in a modified MS Word document; handwritten work will not be accepted. If you
need assistance, please contact your course instructor.
All commonly used formulas for geometric objects are really mathematical models of the characteristics
of physical objects. For example, a basketball, because it is a sphere, can be partially modeled by its
distance from one side through the center (radius, r) and then to the other side by the diameter formula
for a sphere: D = 2r.
For familiar two-dimensional variables length, L, and width, W, the perimeter and area formulas for a
rectangle are mathematical models for distance around the rectangle (perimeter, P) and the region
enclosed by the sides (area, A), respectively:
P = 2L + 2W and A = L x W
Along with another variable, height, H, a three-dimensional rectangular prism’s volume and surface area
can be measured. For example, the formulas for a common closed cardboard box’s inside space
(volume, V) and outside covering (surface area, SA) are respectively:
V = L x W x H and SA = 2(L x W) + 2(W x H) + 2(L x H)
For this Submission Assignment follow Polya’s principles to solve your problems, and include the
following:
 Explain your interpretation of what the problem is about.

 Develop and write down a strategy for solving this problem; show the steps in the correct order
for your attempted solution.
 Did your strategy actually solve the problem? How do you know?
 Suppose your solution did not solve the problem—what would be your next action?
SENDING A PACKAGE
Your goal is to construct a rectangular box with a top on it that has the smallest possible surface area in
which a football and a basketball, both fully inflated, will just fit into at the same time. Pictured below,
the football measures 6.5 inches high and 11.55 inches long, while the basketball is 9.55 inches high:
1. What box dimensions make a good model for this situation? All quantities are inside-of-the-box
measurements. First, position the football and basketball side-by-side. Then, slide the basketball
so that it is even with one point of the football. Now, measurements can be made that will give

the minimum width across both objects. That will be the minimum width of the box with the
smallest surface area. Using the following diagrams, first find the exact LENGTH and HEIGHT. Do
not round:
ANSWERS
Length 11.55 inches
Height 9.55 inches
Explain your answer here: (4 points)
The two objects need to fit into the box in a way that the width of the box is the minimum
possible.
With the documents arranged as shown in the above diagram to fit in the box, the box length
will be equal to the length of the football that is 11.55 inches. The basketball having a length
the of 9.55 inches will fit into the designed box.
Moreover, the height of the box will be equal to the height of the highest item in this case
being basketball. Hence the box height is 9.55 inches. the other item fitted in the sides will
also fit in this box height.
2. Note that the diameters combined include an overlap; see the cross-section perspective below.
To find the WIDTH, you must first account for this by applying the Pythagorean theorem. The
WIDTH will be the radius of the football plus the side b of the right triangle below plus the radius
of the basketball.

Here is the right triangle shown larger and labeled:
Find a and c. The measure of the hypotenuse, c, is the sum of the two balls’ radii. The smaller
side, a, is the difference between these two radii. Find these two exact sides including the units
of measurement. Do not round:
ANSWERS
a 1.525 inches
c 8.025 inches
Explain your answer here: (4 points)
The diameters combined include an overlap that needs to be accounted for before obtaining
the width. To find the box width will apply the Pythagorean theorem to accommodate the
overlap. The radius W of the box will be obtained by the radius of football plus side b of the
right triangle shown above plus the radius of the basketball.
In the right-angled triangle
c=∑ of the radii
=3.25+4.775=8.025 inches

Then a=difference of theradii
That is 4.775−3.25=1.523inches
3. Next, find b. Apply the Pythagorean theorem , a2 +b2=c2, using its form:
b= √c2−a2
Show all step-by-step calculations, including the units of measurement, and round your eventual
answer to the nearest hundredth:
ANSWER
B 8.17 inches
Explain your answer here: (5 points)
b= √ 8.0252 +1.5252
b= √64.400625+2.325625
¿ √ 66.72625=8.168613713
this will be 8.17 inches
4. Now, list all the box’s dimensions in the chart below. Recall from above: The WIDTH will be the
radius of the football plus the radius of the basketball plus the side b of the right triangle above.
Show all step-by-step calculations, including the units of measurement, and round your final
answers to the nearest tenth:
ANSWERS
Length 11.55 inches
Width 16.2 inches
Height 9.55 inches
Explain your answer here: (6 points)

The length and height of the box are obtained from the calculations above.
Now to obtain the width W, will apply the available formula
W =radius of the football+radious of the basketball+ side b
W¿ 3.25 inches+4.775 Inches+¿8.17 inches which is 16.195 inches. This to the nearest tenth
is 16.2 inches.
5. Using Polya’s technique for solving problems, describe and discuss the strategy, steps, formulas,
and procedures you will use to solve this problem.
Explain your answer here: (8 points)
Problem statement
The issue to be resolved is packing a package. The two items (football and basketball) need to
be fitted into a box with the minimum surface area possible.
Strategy
First all take the measurements of the items that is length and height of the items.
Afterwards the items will be arranged in a manner that can make them fit the box properly.
The dimensions of the box are then calculated then its designed and the items packed and
enclosed.
Review
The last step is to ensure that the procedure followed leads to the exact intended outcome.
6. The minimum surface area corresponds to the minimum volume. Using the formula and
dimensions from above, find the box’s volume.

Show all step-by-step calculations, including the units of measurement, and round your final
answers to the nearest whole measurement unit:
ANSWER
Volume 1787 cubic inches
Explain your answer here: (6 points)
The volume V of the enclosed box will be
V =L∗W ∗H
The L=11.55 inches , W=16.2 inches∧H =9.55 inches
Hence V =11.55 inches∗16.2inches∗9.55 inches
This gives 1786.9005 cubic inches. To the nearest whole measurement unit, we have 1787
inches.
7. Using the formula and dimensions from above, find the box’s surface area.
Show all step-by-step calculations, including the units of measurement, and round your final
answers to the nearest whole measurement unit:
ANSWER
Surface area 904 square inches
Explain your answer here: (6 points)
The surface area SA of the closed box is given by
SA=2 ( L∗W ) +2 ( W∗H )+2( L∗H )
the L=11.55 inches ,W =16.2inches , H=9.55inches
Now
L∗W =11.55 inches∗16.2 inches=187.11 square inches
W ∗H=16.2 inches∗9.55inches=154.71 square inches
L∗H =11.55 inches∗9.55 inches=110.3025 square inches

Now the SA
( 2∗8187.11 square inches ) + ( 2∗154.71 square inches ) + ( 2∗110.3025 square inches ) =904.245 square inches
To the nearest whole measurement this gives 904 square inches
8. Demonstrate that your solution is correct. In other words, explain why the box you have created
is the smallest possible box.
Explain your answer here: (7 points)
The box created is the smallest possible. The task was to have a box that can carry both the
football and basketball at the same time. When designing the box, the length was obtained in
such way that the longest object that is the football will touch both the sides of the box with
no inch of space left. On the other hand, the height is such that the height of the highest item
that is the basketball touches the top and the bottom with no inch of space left. The designing
goes ahead to apply the Pythagorean theorem to remove any form of overlap in the width of
the box. With this the box width is such that the two items arranged side by side touches
leaves no room on either side of the width. Being that the box just fit the items exactly leaving
no space at the top or the sides it is proven to be the smallest box possible that can meet our
objective: carrying the football and the basketball.
PAINTING A BEDROOM
The walls and ceiling in your bedroom need to be painted, and the painters’ estimates to do the work
are far too expensive. You decide that you will paint the bedroom yourself. Below is the information to
help you solve the problem:
 The bedroom is 17 feet, 3 inches long by 18 feet wide, and the ceiling is 9 feet high.
 The color of paint you have selected for the walls covers 84 square feet per gallon and costs $31.50
per gallon.
 The inside of the bedroom door is to be painted the same color as the walls.
 The ceiling will be painted with a bright white ceiling paint that costs $27.50 per gallon but only
covers 73 square feet per gallon.
 Two coats of paint will be applied to all painted surfaces.

 The room has one window, measuring 3 feet, 3 inches by 4 feet, which will not be painted.
1. Because different paint lots of the same color may appear slightly different in color, when
painting a room, you should buy all of your paint at one time and intermix the paint from at least
two different cans so that the walls will all be exactly the same color. Because all ending values
are given in feet, first find the room dimensions in feet that make a good model for this
situation. Do not round. (Make sure to use the conversion where 12 inches are in 1 foot.)
ANSWERS
Length 17.25 feet
Width 18 feet
Height 9 feet
Explain your answer here: (5 points)
The length of the room is given as 17 feet 3 inches. Transforming the inches to feet we have
1 foot=12inches
The 3 inches will 3∗1
12 =0.25 foot

The total length is 17+0.25=17.25 feet
The width is 18 feet and the height are 9 feet.
2. Using the measurements found above, label the rectangular sides in feet in this table. Do not
round. (6 points)
SIDE ANSWERS
Left wall or Right wall 18 feet 9 feet
SIDE ANSWERS
Front wall or Back wall 17.25 feet 9 feet
SIDE ANSWERS
Ceiling 17.25 feet 18 feet
SIDE ANSWERS
Window 3.25 feet 4 feet
3. Using the formula concepts and dimensions from above, find the bedroom’s total painted the
surface area around all the walls, including both coats. Do not forget to subtract the window’s
area. Also, double the paint to account for two coats.
Show all step-by-step calculations, including the units of measurement, and round your final
answers to the nearest whole measurement unit:

ANSWER
Total painted wall surface area 1243 square feet
Explain your answer here: (6 points)
The left wall surface area
L∗W =18∗9=162 square feet
The right wall surface area
L∗W =18∗9=162 square feet
Front wall and back wall
SA=2(L∗W )
¿ 2 ( 17.25∗9 ) =310.5 square feet
The window
L+W =3.25∗4=13 feet
Total painted area ¿ front wall+back wall +¿ wall+¿+ceiling−window.
¿ 162+162+310.5−13=621.5 square feet
This will be multiplied by 2 to cover the two coats to give a total of 1243 square foot.
4. Using the formula concepts and dimensions from above, find the ceiling’s total painted surface
area, including both coats.
Show all step-by-step calculations, including the units of measurement, and round your final
answers to the nearest whole measurement unit:
ANSWER
Total painted ceiling surface area 621 square feet
Explain your answer here: (5 points)
The total surface area of the ceiling is
L∗W =1725 feet∗18 feet
¿ 310.5 square feet
To cover the two coats the area is multiplied by 2 to give 621 square feet

5. Describe and discuss the strategy, steps, formulas, and procedures for how you will use Polya’s
problem-solving techniques to determine how much it will cost to paint this bedroom with two
coats of paint (on all walls and the ceiling).
Explain your answer here: (8 points)
Problem explanation
I need to paint the entire ceiling and walls of a room
Strategy
The first step will be to calculate the area of the bedroom to be painted.
Afterwards all gauge the volume of paint that will be enough to paint both the entire ceiling
and the entire wall. Afterwards all evaluate the total cost of the entire volumes of paint
needed. The last step will be to gauge the amount of time that the painting will take.
Review
The last step will involve reevaluating the plan to ensure the steps covers all the elements that
the painting need to consider for it to be completed smoothly.
6. Find, individually and as a total, how much it will cost to paint this bedroom with two coats of
paint (on all walls and the ceiling).
Show all step-by-step calculations, including the units of measurement, and round your final
answers to the nearest whole dollar amount:
ANSWERS
Total cost painted wall surface area $ 466
Total cost ceiling surface area $ 234
Overall total cost of paint $ 700
Explain your answer here: (7 points)
Wall

The total surface area of the wall to be painted is 1243 square feet, one gallon of paint covers
84 square feet. This means a total of 1243
84 =14.79761905 gallons of paint will be used
The paint cost $31.5 per gallon which turns to $ 31.5∗14.79761905=$ 466.125
To the nearest dollar that is $ 466
Ceiling
The total area of the ceiling to be painted is 621 square feet.
One gallon of paint will paint 73 square feet this means 621
73 =8.506849315 gallons will be
used.
A gallon of paint cost $ 27.5 hence a total of $27.5*8.506849315¿ $ 233.9383562
This as a whole number is $ 234
In total the painting process will cost
$ 234+ $ 466=$ 700
7. Assuming you can paint 100 square feet per hour, what will be the work time needed to paint
your bedroom?
Show all step-by-step calculations, including the units of measurement, and round your final
answers to the nearest whole hour amount:
ANSWERS
Total painting time 19hours
Explain your answer here: (7 points)
To paint100 square feet I need one hour, now
The total area to be painted is the area of the ceiling to be painted plus the area of the wall to
be a painted that is
1243 square feet +621 square feet=1864 square feet
To paint the entire area will need 1864
100 =18.64 hours
This to the nearest whole number is 19 hours

References
Costeff, H. (1966). A simple empirical formula for calculating approximate surface area in children. Arch
Dis Child, 681–683.
Rorres, C. (2007). Tomb of Archimedes. Courant Institute of Mathematical Sciences.