Formulas For Geometric Objects And Solving A Packing Problem

1. Explain your interpretation of what the problem is about.

2. Develop and write down a strategy for solving this problem; show the steps in the correct order for your attempted solution.

3. Did your strategy actually solve the problem? How do you know?

4. Suppose your solution did not solve the problem—what would be your next action?

Geometric Formulas for Common Objects

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? 

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:

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

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  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.

Solving a Packing Problem Using Polya's Principles

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:

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

=3.25+4.775=8.025 inches

Then

That is

Next, find b. Apply the Pythagorean theorem,using its form:

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:

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:

The length and height of the box are obtained from the calculations above.

Now to obtain the width W, will apply the available formula

W8.17 inches which is 16.195 inches. This to the nearest tenth is 16.2 inches.

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 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. 

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. 

The last step is to ensure that the procedure followed leads to the exact intended outcome.

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

Calculating the Surface Area and Volume of the Box

Explain your answer here: (6 points)

The volume V of the enclosed box will be

The

Hence

This gives 1786.9005 cubic inches. To the nearest whole measurement unit, we have 1787 inches.

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

Now

Now the SA

To the nearest whole measurement this gives 904 square inches

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:

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.  

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: 

The length of the room is given as 17 feet 3 inches. Transforming the inches to feet we have

The 3 inches will

The total length is 17

The width is 18 feet and the height are 9 feet.

Using the measurements found above, label the rectangular sides in feet in this table. Do not round.

	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

Verifying the Solution

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:

The left wall surface area

The right wall surface area

Front wall and back wall

The window

Total painted area.

This will be multiplied by 2 to cover the two coats to give a total of 1243 square foot.

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:

The total surface area of the ceiling is

To cover the two coats the area is multiplied by 2 to give 621 square feet

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:

Problem explanation

I need to paint the entire ceiling and walls of a room

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.

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.

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: 

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

The paint cost $31.5 per gallon which turns to

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  will be used.

 A gallon of paint cost $ 27.5 hence a total of $27.5*8.506849315

This as a whole number is $ 234

In total the painting process will cost

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:

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

To paint the entire area will need

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.