Three Acts: Toilet Roll (101questions)

ACT 1 – Any questions?

(Uploaded onto 101questions)

ACT 2 – What information would I need to know to answer this question?

The Dimensions of the toilet paper. All measurements in millimetres (mm).

Lower and Upper Bounds for Thickness, T (LB – 0.05mm; UB – 0.17 mm)

Lower and Upper Bounds for inner diameter, d (LB – 40.99mm; UB – 42.41mm)

Lower and Upper Bounds for full diameter, D (LB – 108.02mm; UB – 112.03mm)

The measurements were taken using a digital caliper. Due to the texture of the material, there are inherent problems with these measurements. I was only able to get estimated lower bounds and upper bounds of each dimension.

ACT 3 – the answer.


Method 1 – One way of calculating the length of toilet paper is to calculate the volume of the toilet paper when it is has been rolled out and the volume when it is unrolled and put them equal to eachother.

Volume of Rolled out Toilet Paper (Cuboid) = Volume of unrolled out toilet paper

Method 2 – Another way to calculate the length would be to consider the toliet paper as a 2-dimensional object (you could do this in the method above as well and not have to worry about the width) and add up all of the layers. By adding up all of the layers, you get the cross-sectional area of the toilet roll. The easiest way to add them all up is to integrate.

Cross sectional area,

The cross-sectional area is also equal to the Length x Thickness,

Equating the areas we get back to the equation derived in Method 1.

Alternatively, we could simply add up each layer seperately.

Using the summation formula for an arithmetic series, we can reduce this to,

Since n represents the number of layers,

Finally, substituting this formula into the previous one, we get back to the original formula.

Now that we have an equation, we can calculate an upper and lower bound for the length.

For the Upper Bound, we need to use the upper bound for the full diameter, the lower bound for the inner diameter and the lower bound for the thickness and vice versa for the Lower Bound.

Since the actual length of the toilet paper is 30.98m, the mathematics did not help in the slightest in this case; measurement errors were just too big, especially regarding the thickness of the toilet roll. Having said this, the project promoted awesome discussion at every stage.

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4 Responses to Three Acts: Toilet Roll (101questions)

  1. Bruno Reddy says:

    Love the 3 Acts approach. I’ve not taken the plunge yet.
    It’s a great post too with the calculations, etc. Thanks for sharing it Dan.

  2. Thanks for the comment Bruno, it was great fun doing it! I’d like to get a few more going once I’ve settled in at my new school. Hopefully try to take it a step further by getting the students to have a go at making a first act.

  3. hai, thanks for share this information. but I have a question about n. n is the number of layer which mean the number of turns for a roll of toilet paper or the number of sheet?? can you help me explain this. thanks

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