Abstraction is a major driving force of progress. By finding solutions that are more and more general, we can increase the scope of our practical and theoretical knowledge.
It is from this perspective that I have approached Glen's impressive work at Agoraphilia. When it looks like half of a roll of toilet paper is left (based on the change in the radius of the roll), Glen asked, how much of the tp is really left? Using measurements he took of a typical roll, (4 cm in tube diameter and 10 cm in total diameter), Glen found that the answer was about 39% (or, more precisely, 11/28). He then generalized his result to the case where the apparent proportion of roll used is a variable, X. Then the actual fraction of remaining toilet paper, Y, is
Y = [(3X + 2)^2 – 4]/21
In the interest of peer review and further generalization, I treated both the radius of the tube, r, and the radius of the entire roll, R, as variables. Then, using only geometry (calculations of the volumes of cylinders), Y can be found to equal:
Y = [(X^2)(R - r) + 2Xr]/(R + r)
Though it isn't obvious, this formula simplifies to his formula when R = 5 and r = 2. It also verifies Glen's claim that his formula is correct when both r and R change proportionally, since both the numerator and denominator are linear in r and R. That also means that it doesn't matter if you use centimeters or inches or some other unit, or if you measure the diameter or the radius, as long as you're consistent in your choices.
As a test case of this formula, I calculated Y for a roll with a 10" diameter and a 1.625" diameter core, which fits the Surface Mounted Jumbo - Roll Toilet Tissue Dispenser. In that case, when it looks like half of the roll is gone, in reality only [(.5^2)(10 - 1.625) + 2(.5)(1.625)]/(10 + 1.625) = 32% remains. And when it looks like 10% of the roll is still left, there's actually only 3.5% of a roll on the tube.
Chalk another one up for progress.
Update: The fact that changing r and R proportionally does not change Y means that the general formula for Y can be simplified further. Let T be the the ratio of the total diameter of the toilet paper roll to the diameter of the tube, T = R/r. Then
Y = [2X + (T - 1)X^2]/(T + 1)