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I came across a puzzling problem today (or, at least, one aspect of it was puzzling):

Imagine you have a roll of tape and you wrap it around a cylinder of radius 3: The outer radius of a roll containing 20 metres of tape is 4 cm. Approximately, what is the outer radius of a roll containing 80 metres of tape?

A. $5$

B-$5.5$

C-$6$

D-$7$

E-$12$

What I know so far

  • The cross sectional area of a roll of tape with radius $4$ is $(4^2 - 3^2)\pi \tag{1}$

What I need help with:

The very brief answer solution given says that because of fact (1):

  • The cross section of the $80$m tape is $28 \pi \tag{2}$
  • Thus the outer radius is approximately $6$ (C)

Please clarify:

  • Exactly what is meant why 'outer radius'?
  • Why does fact (1) directly imply fact (2)? Isn't the area scale factor squared? So if the length ratio is $80/20 =4$, the area ratio should be $16$?
  • How did they get to the final answer of 'approximately $6$'?

Many thanks!

Edit - I understand that my question is a variation of math.stackexchange.com/q/1633704/399263, However, rather than trying to find the length, knowing the radius, I'm trying to find the radius, knowing the length. Whilst I appreciate that the technique should be the same, I'm struggling to understand why certain things are (for example the relationship between the area and the length, explained below). I would really appreciate it is someone could take 2 minutes to explain the link to me please.

vgupt
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    For exact calculation see https://math.stackexchange.com/q/1633704/399263, CiaPan answer could be used for quick approximation. – zwim Oct 08 '20 at 18:30
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    Does this answer your question? Calculating the length of the paper on a toilet paper roll – Mark S. Oct 08 '20 at 20:11
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    Firstly, thanks @zwim for taking the time to read my question. The link you have posted is very useful and I had a read through some of the material discussed. A lot of it went over my head, but I think CiaPan's answer is getting at the heart of my question. I still don't follow, however, - why is the area between circles with radii and r equal to the length times the thickness of one sheet? And going back to my original question - if the length increases from 20 to 80 (x4) should the area increase by a factor of 16? – vgupt Oct 09 '20 at 06:48
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    It only allowed me to tag one user, but thank you @MarkS. as well. Further comments above. – vgupt Oct 09 '20 at 06:49
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    @zwim - please could you help me? Hadn't heard back so thought I'd ask again... – vgupt Oct 10 '20 at 10:05

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