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Calculate the diameter of a roll if a paper with thickness 0.14mm and 1100m is wrap around the roll, given the roll have a diameter of 92.2mm.

For this question i tried two method Consider total surface area wrap on the roll=length*paper thickness While the roll have surface area of

$\pi (R_{roll}^2-R_{core}^2)=1100000*0.14$

which get diameter of the roll=$452.302mm$

While 2nd method consider cutting the paper with $2\pi r$ length to be wrap around the roll which constitute the 1st layer. While next layer will need $2{\pi (r+0.14)}$,3rd layer will need $2\pi (r+2(0.14))$ and so on.

which can be compute as $2\pi r + 2\pi (r+0.14) + 2\pi (r+0.28)+....=1100000$

$n(2\pi r)+2\pi (\frac{n}{2}((2)(0)+ (n-1)(0.14)))=1100000$

$n=1286$

which mean at 1286 layer, the paper will fully wrap around the roll and hence getting total thickness =1286x0.14=180.04mm

Total radius=core radius+radius of paper wrap around core=180.04+46.1=226.14mm

Diameter=226.14mmx2=452.28mm

Both consideration give value quite close to each other but mathematically which one is more accurate?

David G. Stork
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chuackt
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    please check this link where the similar issue have been researched https://math.stackexchange.com/questions/1633704/the-length-of-toilet-roll – Vasili Aug 31 '18 at 18:04

1 Answers1

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Note that the difference is well less than the thickness of one wrap. I suspect the difference comes from the fact that the last wrap is not complete. The first answer averages the thickness of the final wrap-if the wrap were $\frac 13$ way around it would give you $\frac 13$ of a sheet thickness added to the radius. The second counts whole thicknesses. Your sum for $1286$ turns comes out slightly less than $1100$ meters.

Note that your input data is only good to two places, so your output should only have two places as well. Your two approaches agree much better than this. If you take the input data as exact you need to model more carefully what happens when the wrap comes to the start of the next layer. You have assumed a sharp stairstep, which is not realistic. Even more, you have assumed that the paper is wrapped perfectly tightly around the roll. I suspect the roll will be somewhat larger than you calculate do to the wrap not being tight.

Ross Millikan
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