Kitchen Math: What is the density of paper towel sheets?

Kitchen math is important math for food service and catering. Gastronomy requires both math and measurement. The related kitchen duties also require some understanding of the physical properties of equipment and supplies.

Costco Paper Towel / Courtesy: Costco Wholesale
Costco Paper Towel / Courtesy: Costco Wholesale

In the kitchen, paper towels are useful for food preparation and essential for food safety. Like everything in the food service industry, paper towels are resources that must be managed to help control costs.

Food service managers use comparison shopping to find the best deals. For paper towels, density and absorbency are very important product features.

In this math practice activity, we we will focus on the density of paper towel sheets.

TL;DR
The density of the paper towel is 0.00784 pounds per cubic inch in this specific math solution. That is roughly equal to 0.2 grams per cubic centimeter.

How do you find the density of paper towel sheets?

The answer depends on the quality of the paper towel sheets. Our test sample for this math activity is part of the package called Kirkland Signature Big Roll Paper Towels from Costco Wholesale.

paper towel rolls
Paper towels / Courtesy: Costco Wholesale

Let us first review the governing equation for density.

Density = Weight / Volume

Density is a property that tells us how closely-packed molecules are within a unit volume. Higher densities tend to be heavier, and lower densities tend to be lighter.

Subways during rush hour are densely-packed with passengers and are heavier. During off-hours, the passenger load is much lighter. The same principle applies for molecules.

We already know that matter is anything that has mass and takes up space. Density helps us better understand how much matter is packed into a specified three-dimensional space.

To achieve that understanding, we need to know the amount of mass, which is affected by gravity (which causes the phenomenon called weight) and the amount of volume.

Units for density are weight per unit volume. For this example, we will specifically use the units pounds per cubic inch [lbs. / cu. in.]

The weight of the paper towel material is unknown, so we will use a digital kitchen scale to take some measurements.

Digital Kitchen Scale / Courtesy: Amazon
Digital Kitchen Scale / Courtesy: Amazon

We find that the total weight of an unopened, unused paper towel roll with its packaging is 12.4 ounces.

The cardboard roll that holds the paper towel is 0.2 ounces. The packaging that wraps the paper towel roll is also 0.2 ounces.

In other words, we can calculate the weight of the paper towel material itself by subtracting the weight of everything else (e.g., all that other stuff).

W_material = W_total – W_stuff
W_material = 12.4 – 0.4

The paper towel material weighs 12 ounces. There are 16 ounces in one pound. So, we can say that material weight is 12/16 pounds.

Ask your voice assistant:
"how many pounds is 12 ounces?"

We have calculated the weight of the paper towel material. Now, let us calculate the volume of the paper towel material.

We know that the area of the entire paper towel roll is 85 square feet. We can calculate the area A (in square inches) to be:

A = 85 ft. ft. * 12 * 12 in./ft. in./ft. ==> 12,240 in. in.

In other words, we simply converted the area from square inches into square feet. All the voice assistants can readily convert common units.

Ask your voice assistant:
"85 square feet in square inches"

CCSS.MATH.CONTENT.5.MD.A.1: Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

The paper has a thickness, but a ruler won’t be practical for this measurement. We need something that can measure very small fractions of an inch.

Let us use digital calipers.

Digital calipers
Digital calipers / Courtesy: Amazon

Using digital calipers, we measure that the thickness t (in inches) of a paper towel sheet is 1/128″.

We now have enough information to calculate the volume of the paper towel material. The volume V (in cubic inches) is:

V = A * t
V = ( 85 * 12 * 12 ) * 1/128

The weight W = 12/16 lbs. Combining all our previous calculations, we find that the density must be:

ρ = W / V = (12/16) / ( 85 * 12 * 12 * (1 / 128))

The density of the paper towel sheets is 0.00784 lbs. / cu. in.

All the calculations above can be organized into a computational solution that can be solved using Wolfram Alpha.

Sample Wolfram Alpha input
W=(12/16);V=(85*12*12*(1/128));rho=W/V
View Result

Conclusion

This information can be recorded and used later for comparison shopping of paper towels that may enter the market at a future date (hint, hint). While price may be an important factor, the density of the paper towel sheets for cooking and cleaning is also a very important metric in food service.

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