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roll of toilet paper, ratio of length of paper to diameter of roll?

  • 01-02-2009 2:49pm
    #1
    JoeB-
    Registered Users, Registered Users 2 Posts: 689 ✭✭✭


    Hi

    Can anyone give a ratio between the length of a toilet roll for example, and the diameter of the final roll?

    For example, we'll say that the outer diameter of the inner cardboard core is 30mm... and the paper is 1mm thick... and so the diameter after one complete turn is 32mm (two thicknesses of paper).

    It's tricky as there are two ways of looking at it.. the first is to say that the paper rolls like a spiral... so the radius of the roll is constantly changing, sounds like calculus is needed.

    But in fact, when we consider the very start of the rolling, we can see that the radius stays the same for the first circumference, then it steps up to a new wider circumference, and doesn't increase again until it has completed a new circumference, then it steps up again. It might be that this could be solved by a series, or an arithmetic or geometric progression.. (maybe, or maybe the terms change in a way that meakes this impossible)

    Anyway, using the figures above, (30mm outer diameter inner core, 1mm thick paper)... can anyone give the diameter of the finished roll when we roll on 10 meters of paper.?
    (And two answers maybe, if it is rolled as a spiral, and if it rolled as a series of steps?)

    Also how many layers of paper as well I suppose... :P (i.e. during what roll will the 10meters be finished on, the 3rd, 21st, 105th etc)

    I think this is tricky enough...
    I reckon the 10meters will be rolled on about 3/4 way through the 43rd layer, if it is stepped on... I will give my easy method later, if anyone reads this forum. The diameter of the finished roll would be 115.5mm


Comments

  • #2
    [Deleted User]
    Posts: 4,630 ✭✭✭ [Deleted User]


    It's an interesting question. Something very similar was dealt with here a few weeks ago, actually.


  • #3
    JoeB-
    Registered Users, Registered Users 2 Posts: 689 ✭✭✭JoeB-


    Yes, it is similar.

    The cross sectional area is a good approach.. it isn't 100% accurate as it assumes the final roll is exactly circular, and not a spiral, or a step which it will be in real life.

    By my calculations the paper has a cross section of 10,000 x 1 = 10,000.. the 30mm core has a cross section of 706.86mm, and a circle with a cross section of 10,706.86 (area) has a radius of 58.38mm.. up from a radius of 15mm originally (the core).. gives 43.38 as the increased radius, or 43.38 complete turns on the roll.

    Very good, but not 100% accurate...


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