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A standard roll of paper towels consists of a cardboard tube with outer diameter $4$ cm.

Imagine the paper being wound onto the cardboard tube.

After each complete winding, the total diameter of the roll increases by an amount $2t$, where $t$ cm is the thickness of the paper.

Let $S_n$ denote the total length in cm of paper wrapped around the tube when it is wrapped around $n$ times, so that $S_0 = 0$.

Write a recursive relationship that expresses $S_n$ in terms of $S_{n-1}$.

Which do you think is correct: $S_n = S_{n-1}+\pi(4+n2t)$ or $S_n = S_{n-1}+\pi(4+(n-1)2t)$?

In other words, after one winding, would the length of paper around the tube be $(4+2t)\pi$ or $4\pi$?

barak manos
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1 Answers1

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$t$ has to be small for this to make sense, so it won't matter much. If you think about the first layer, the inner diameter is $4$ and the outer diameter is $4+2t$. The inner and outer circumferences are $\pi$ times these. I think the most reasonable approach is to average them, getting an effective diameter of the first layer as $4+t$, then increasing by $2t$ each additional layer.

Ross Millikan
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