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.