Mathematical Modeling and Problem Solving

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:

POLYA’S PRINCIPLE STEP 1: UNDERSTAND THE PROBLEM

1. Describe, in detail, your interpretation of what you understand the problem to be. In other words, what problem will you need to solve? Is there enough information to enable you to find a solution to your problem? Explain your answer here: (6 points)

2. 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.

As a guide, use the given model diagrams and the sketch of the box to first find the exact LENGTH and HEIGHT.

Next, let’s assume the WIDTH will be the diameter of the football plus the diameter of the basketball; see the cross-section perspective, shown outlined here in blue:

Now, list all of the box’s dimensions in the chart here:

Show all step-by-step calculations, including the units of measurement. Do not round.

Explain your answer here: (4 points)

POLYA’S PRINCIPLE STEP 2: DEVISE A PLAN

3. Using Polya’s technique for problem solving, describe in detail, what your plan will be to solve this problem. In other words, what is your solution strategy? Discuss the strategy, steps, formulas, and procedures you will use to answer this problem.

Explain your answer here: (8 points)

POLYA’S PRINCIPLE STEP 3: CARRY OUT THE PLAN

4. 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:

Explain your answer here: (4 points)

5. 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:

Explain your answer here: (6 points)

POLYA’S PRINCIPLE STEP 4: TAKE A LOOK BACK

6. Did this strategy actually solve the problem? How do you know? Demonstrate that your solution is correct. In other words, explain why the box you created is the smallest possible box. Was this the best way to solve this problem? If you had to do this again, what would you do differently? What would you do the same?

Explain your answer here: (6 points)

Continued below…

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.

· 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.

POLYA’S PRINCIPLE STEP 1: UNDERSTAND THE PROBLEM

1. Describe, in detail, your interpretation of what you understand the problem to be. In other words, what problem will you need to solve? Is there enough information to enable you to find a solution to your problem?

Explain your answer here: (6 points)

2. Begin to list the facts you know. 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.) Use the sketch of the room as a guide.

Explain your answer here: (4 points)

3. Using the measurements found above, label the rectangular sides in feet in this table. Do not round. (6 points)

Diagram 2

Wall

14.25’ by 8’

POLYA’S PRINCIPLE STEP 2: DEVISE A PLAN

4. Using Polya’s technique for problem solving, describe in detail, what your plan will be to solve this problem. In other words, what is your solution strategy? Discuss the strategy, steps, formulas, and procedures you will use to answer this problem.

Explain your answer here: (8 points)

POLYA’S PRINCIPLE STEP 3: CARRY OUT THE PLAN

5. Using the formula concepts and dimensions from above, find the bedroom’s total painted surface area around all of the walls. Do not forget to subtract the window’s area. Also, determine the amount for two coats by doubling the paint previously found.

Show all step-by-step calculations, including the units of measurement, and round your final answers to the nearest whole measurement unit:

Explain your answer here: (6 points)

6. 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:

Explain your answer here: (6 points)

7. 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:

Explain your answer here: (8 points)

8. 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:

Explain your answer here: (6 points)

POLYA’S PRINCIPLE STEP 4: TAKE A LOOK BACK

9. Did this strategy actually solve the problem? How do you know? Demonstrate that your solution is correct. In other words, explain why the values you have created are the best times and dollar amounts for the job. Was this the best way to solve this problem? If you had to do this again, what would you do differently? What would you do the same?

Explain your answer here: (6 points)