Question

MATH125: Unit 1 Individual Project 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. Submit this modified Answer Form in the Unit 1 IP Submissions area.
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 IP assignment follow Polya’s principles to solve your chosen problem, 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. The following are the measurements of the football and Mathematical                                    
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:


ANSWERS
LENGTH 11.55 inches
HEIGHT 9.55 inches
6
Explain your answer here:


Football is the longest item I got the from the football which 11.55”
And the has the height which is 9.55”







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 of these two radii. Find these two exact sides including the units of measurement:
ANSWERS
a                               2.0”
c 42.2”

Explain your answer here:     a = 11.55 – 9.55 = 2.0 the difference of the two radii.
                                                    c = 11.55(2) + 9.55(2) = 84.40 the units of measure




Next, find b. Apply the Pythagorean theorem,a^2+b^2=c^2, using its form:
b=√(c^2-a^2 )
Show all step-by-step calculations, including the units of measurement, and round your final answer to the nearest hundredth:
ANSWER
b 4.48

Explain your answer here:
C = 11.55(2) + 9.55(2) = 42.20(2)
b = the square root of C to the second power minus a to the second power



Now, list all of 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
WIDTH
HEIGHT

Explain your answer here:



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:




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

Explain your answer here:





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

Explain your answer here:






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:

Answer 1