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Write-up


Title
Postal Restrictions
Problem Statement
The U.S. Postal Service will only mail packages that meet certain size requirements. For cylinder-shaped packages (or "rolls"), the minimum length is 4 inches and the maximum length is 36 inches. There is also a restriction that the length plus two diameters can be no more than 42 inches (why do you think they have this restriction?).

What are the dimensions of an acceptable cylinder-shaped package that will have the greatest volume?



Problem setup

This investigation wants to know what is the cylinder with the greatest volume that meets the following two conditions:

 

1)      The length must be between 4 and 36

2)      The length plus two diameters must be less than 42

 

Plans to Solve/Investigate the Problem

I will use Excel to solve this problem. I intend to enter the length and diameter, and use formulas to calculate the volume. I will also have a column which calculates the length plus two diameters. I will limit my lengths to all integers between 4 and 36, and will calculate for all diameters from 1 to n where n s the first integer that the length of 4 exceeds our length + 2*diameter requirement.

 

Investigation/Exploration of the Problem

I created the excel spreadsheets as detailed above. I then found the height and diameter that created the greatest volume. Below is a copy of the cells and columns that returned the greatest volume. You may click the link below to view the entire spreadsheet.

 

EXCEL SPREADSHEET

 

diameter = 14

 

 

 

Length

diameter

volume

length + 2d

(min - 4)

 

 

(<42)

 

(max - 36)

 

 

 

 

4

14

615.7522

32

 

5

14

769.6902

33

 

6

14

923.6282

34

 

7

14

1077.566

35

 

8

14

1231.504

36

 

9

14

1385.442

37

 

10

14

1539.38

38

 

11

14

1693.318

39

 

12

14

1847.256

40

 

13

14

2001.195

41

 

14

14

2155.133

42

 

15

14

2309.071

43

************

16

14

2463.009

44

************

17

14

2616.947

45

************

18

14

2770.885

46

************

19

14

2924.823

47

************

20

14

3078.761

48

************

21

14

3232.699

49

************

22

14

3386.637

50

************

23

14

3540.575

51

************

24

14

3694.513

52

************

25

14

3848.451

53

************

26

14

4002.389

54

************

27

14

4156.327

55

************

28

14

4310.265

56

************

29

14

4464.203

57

************

30

14

4618.141

58

************

31

14

4772.079

59

************

32

14

4926.017

60

************

33

14

5079.955

61

************

34

14

5233.893

62

************

35

14

5387.831

63

************

36

14

5541.769

64

************

 

 

 

 

Extensions of the Problem

Possible extensions would be to examine the maximum volumes for other 3-D shapes, such as triangular prisms and rectangular prisms. You could use excel to find the greatest volume values in the same manner that you did for the cylinder. You would have to look up the postal regulations for each of the different shapes.

Author & Contact
Jim Taylor
jtaylor1@rockdale.k12.ga.us

Link(s) to resources, references, lesson plans, and/or other materials
Link 1
Link 2

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