Loo Roll Emergency

You must have noticed that rolls of toilet paper look healthily full for ages and ages, then seem to run out all of a sudden when you’re alone in the house…

The thing to do, of course, is to make sure there’s a spare roll nearby whenever the current roll is getting a bit low – but how can you tell that, simply from the thickness of the remaining roll?

The Problem: 

A typical new roll of toilet paper is 12cm in diameter, of which the central cardboard tube makes up the middle 4cm. 


Assuming the paper doesn’t get more squashed at either the outer or inner sections, you can assume that the length of paper remaining is proportional to the volume of paper remaining, and hence to the cross-sectional area remaining...


If Desperate Dan’s loo roll has got down to half its original width (i.e. diameter of 6cm), what percentage of paper is actually left?


What is the diameter when half of the paper has been used?


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Last modified: June 18, 2007