Eventmath/Lesson plans/Dimensional analysis, shipping, and an impossible weight limit

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Lesson plan overview
Title	Dimensional analysis, shipping, and an impossible weight limit
Assumed knowledge	Some familiarity with mixed numbers, decimals, units, and multiplication/division. The volume formula for a box. Optional: spreadsheet formulas.
Activities	Students will read a tweet about how it is impossible to exceed the weight limit of a flat rate box. Students will then do a multi-step activity. Step 1: compute the volume of the box. Step 2: Use the density of several common objects, to compute the mass of the box when filled completely with that object (neglecting the weight of the box itself). Step 3: compare to the mass limit, and draw conclusions.
Class time	00-15 minutes
Source
"Paul Sherman: It is physically impossible to exceed the 70-pound domestic weight limit for a small flat rate box". Twitter. 2022-04-20. Archived from the original on 2022-08-18.
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Activities

Given information

• Box dimensions: Length = 8 5/8 in, Width = 5 3/8 in, Depth = 1 5/8 in.
• Domestic Weight limit: 70 lbs

Step 1: compute the volume of the box

Formula: Length x Width x Height = Volume (the units multiply too)

Beware: Students seem to pay more attention to the USPS image with "8.5 in x 5.5 in x 1.5 in" which is not the same. It is clear that Paul Sherman used the exact values, and the students should too if they are to compare their results to his.

Step 2: Use the density of common objects to obtain the mass of a full box (neglecting the cardboard)

Formula: Volume x Density = Mass (metric optional).

Common densities

Material	g/cm3	lb/in3
Pine	0.5	0.018
Book paper	0.72	0.026
Granite	2.7	0.097
Steel	7.8	0.28
Gold	19.3	0.69
Osmium	22.5	0.81

Avoiding metric may lower the bar, but in real life a person might need to be able to convert units (densities found online are commonly listed in metric). Even if metric is avoided, the calculation should be done with units, and involve the cancellation of in^3

Extra details for metric

Basic conversions

• 1 in = 2.54 cm
• 1 lb = 453.59 g

The conversions can be done in more than one way, for example:

1. The volume could be converted to cm³, then multiplied by the density to get the mass in grams, then converted to pounds and compared with 70.
2. The densities could be converted from g/cm³ to lb/in³, then multiplied by the volume.

Sample conversion (needed in the second approach) $1{\frac {g}{cm^{3}}}=1{\frac {g}{cm^{3}}}\cdot {\frac {1lb}{453.59g}}\cdot {\frac {2.54^{3}cm^{3}}{1^{3}in^{3}}}={\frac {2.54^{3}}{453.59}}{\frac {lb}{in^{3}}}\approx 0.036{\frac {lb}{in^{3}}}$

Step 2a (optional) Estimate the error

Have the students redo their calculations with both the density and the volume rounded up by 1 in the last place.

Step 3 Compare and draw conclusions

Note: Many students may say "No" to the first question below. The values are highly sensitive to rounding error. His values are accurate enough for his conclusions to be correct. Step 2a is intended to illustrate the fact that even rounding error will not cause the conclusions to be incorrect.

"Questions"

• Are the numerical values posted by Paul Sherman correct?
• What might be a reasonable weight limit?
• Would including the mass of the cardboard box itself change the answer significantly? Why?

Assignments

You're welcome to suggest exercises, activities, assignments, or projects based on the material of this lesson.

Resources

USPS Flat Rate Small Box https://store.usps.com/store/product/shipping-supplies/priority-mail-flat-rate-small-box-P_SMALL_FRB

Densities:

• https://www.lenntech.com/periodic/elements/os.htm
• https://www.thoughtco.com/table-of-densities-of-common-substances-603976
• https://cedarstripkayak.wordpress.com/lumber-selection/162-2/
• https://www.paperonweb.com/density.htm
• https://hypertextbook.com/facts/2004/ShayeStorm.shtml

Background

Volume

• https://www.youtube.com/watch?v=RxkRlIAucMk

Fractions

• https://www.youtube.com/watch?v=alstJ37BoZo
• https://www.youtube.com/watch?v=bf9GSi60Hr8
• https://www.khanacademy.org/math/arithmetic/fraction-arithmetic

Decimals

• https://www.youtube.com/watch?v=2xGzQXn3WUQ
• https://www.khanacademy.org/math/arithmetic/arith-decimals

Explorations

You're welcome to share references for additional learning and exploration, such as links to other articles, videos, spreadsheets, or computer code. When an open-access substitute is unavailable, links to paywalled sites are acceptable in this section.

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