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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.

 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

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