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October 1, 2007
New York, N.Y.
A couple years ago a friend of mine got a New York City Teaching Fellowship to teach in the public schools here. The subject she chose to teach was mathematics because, as she said, "In math there's only one correct answer."
That statement came to mind while I was sitting in deep contemplation pondering a package of Charmin, a product that Procter & Gamble discreetly refers to as "bathroom tissue" but which in our genteel household goes by the name papier de twa-lay, as in "Hey, who left one square of papier de twa-lay on the roll?"
Charmin comes in four distinct roll sizes, called Regular, Big, Giant, and Mega. No, these sizes do not accomodate four different sizes of bums. The Big roll has twice as many sheets as the Regular, the Giant has 2.5 times as many sheets as Regular, and Mega has four times as many, as shown on this convenient chart:
The Mega roll is so big that it generally requires an adapter to fit in the typical puny papier de twa-lay dispenser.
Each sheet on the roll, we are informed, is 10.8 cm (4.27") wide — that's the width of the roll itself — and 10.1 cm (4") long. So, the Regular roll is a total of 176 × 4" long, or 58-2/3 feet. From there you can easily determine the lengths of the other four sizes: 117-1/3 feet, 146-2/3 feet, and 234-2/3 feet.
These numbers refer to the total length of the roll as it might appear when draped over a tree branch on Hallowe'en. But that's the trivial calculation. If you're the type of person who gazes off distractedly and thinks "I wonder what the relationships are between the radii of these four sizes" then you just might be a mathematician.
This particular problem has an easy solution and a hard solution, and neither approach involves anything so crass as actually measuring the things with a ruler. Even the hard way involves a simplification, but it all turns out to be OK in the end.
I'll be using subscripts R, B, G, and M to refer to the Regular, Big, Giant and Mega sizes. The relationships between the Length of each roll is available from the chart:
Let's give ourselves the freedom to refer to L_{i} where i can be R, B, G, or M. Let's define F_{i} as the multiplicative Factor of the lengths relative to L_{R}. Hence,
For i equal to R, B, G, and M, F_{i} equals 1, 2, 2.5, and 4, respectively.
Let's also define variables for the radii:
We can also refer to R_{i} where i equals R, B, G, and M (but not T). We're basically interested in finding R_{i} as a function of R_{R}, R_{T}, and F_{i}.
A couple other symbols will be useful:
Of course, we have to assume that the paper on the four sizes of rolls is wound around the paper tube consistently and uniformly among the roll sizes. The Volume of the paper based on the Length, Width, and Paper thickness can be calculated like so:
Because each size roll has the same Width and Paper thickness, the Volume is obviously proportional to the Length or,
The Volume occupied by the paper can also be calculated as the volume of a cylinder, which is the area of the cross section of the cylinder (πR^{2}) times the Width of the roll. It's complicated a bit because the volume occupied by the paper tube must be excluded:
For the Regular roll, the volume is:
These two equations can be hooked together using the relationship between the two volumes shown above:
All the W terms drop out because we really could have performed this exercise by just examining the cross section. The π's drop out as well:
So,
And here's the final result that gives us the desired relationship:
Anyone who cares to measure the radius of the paper tube and the Regular roll can plug the numbers in and determine the radii for the larger rolls. For example, if R_{T} (the radius of the paper tube) is 0.75" and R_{R} (the radius of the Regular roll) is 2" (and I'm pretty much making up these numbers because I'm not going to measure them) then the radius of the Big roll would be 2.72".
The more difficult approach to this problem is to treat the roll of paper as a series of concentric circles of paper:
The paper on the roll is actually a spiral, but obviously these concentric circles are a good approximation. Each size of Charmin is associated with a characteristic number N_{i} (where i equals R, B, G, or M) that indicates the number of concentric circles of paper in the roll.
These concentric circles of paper always have a thickness of P with an inner side and an outer side. The inside of the innermost circle of paper is at radius R_{T}, the radius of the tube. The outer side is at radius R_{T} + P. The average is the radius R_{T} + P/2. The next circle of paper has an average radius of R_{T} + 3P/2, and so forth. In general, for circle n (where n ranges from 1 to N_{i}), the radius is R_{T} + (2n-1)P/2. Thus, each of these circles of paper has a circumference equal to:
The entire length of the paper on the roll is equal to the summation of these circumferences (and forgive the slight change in appearance as I switch to bitmaps of an MS Word document created using the Equation Editor):
This doesn't look like it's going to come anywhere close to the original result, but let's move on anyway. We can pull some constant stuff outside the summations and break the summations down:
And then we can reduce it even further to leave just one summation:
The sum 1 + 2 + 3 + ... + K is always equal to (K)(K + 1) / 2, so:
We need to get rid of the N_{i} term if there's any hope of making this method match up with the first one. Fortunately, it's possible to calculate R_{i} knowing the characteristic number N_{i}. Just multiply it by the thickness of paper P and add R_{T}:
From that, solve for N_{i}:
Now substitute that into the formula for L_{i}:
This is just getting worse and worse, but let's expand that square in the first expression and also expand the second expression:
Fortunately, stuff starts falling out, until you're left with:
For the Regular roll,
Since
then
which is exactly one of the intermediate results I got in the earlier derivation.
It's math: There's only one right answer, and the two methods are equivalent.
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(c) Copyright Charles Petzold
www.charlespetzold.com
Comments:
So that's what you were doing for so long.
— Deirdre, Mon, 1 Oct 2007 12:44:44 -0400 (EDT)
Yeah...my eyes glazed over after you defined W and P. What's sad is Math used to be my favorite subject...until I met Discrete Math :P
— Michael Brown, Tue, 2 Oct 2007 13:27:42 -0400 (EDT)
Forgive the dyslexic email addr. I am sure you can manage.
Paper length L, thickness t draped over tree, or laid along street, viewed edge on has area Area = L*t
When wound up on cylinder radius r, ending up big roll radius R, viewed from side, has area
Area = pi*R*R - pi*r*r
So L*t = pi(R*R - r*r)
I just *know* you can figure the final diameter from R, allowing the roll has not swelled from being stood on end in the wet :) for me, sums of series was to much like hard sums!
The value of 't' is a notional equivalent to any real measured value of t, it being a constant for the bogroll, and discovered by an unwinding experiment that is *always* interpreted as vandalistic mischief when undertaken on the pretty pedestrian suspension bridge between the Royal Festival Hall and Embankment station, overlooked by the 'London Eye' Ferris wheel gondolas. (Try 'London Eye' into Google Earth)
My regards
Graham
— graham, Thu, 25 Oct 2007 18:18:01 -0400 (EDT)
I want to calculate longitudinal length of paper roll,I have one increamental encoder for paper length measuring.Paper is rolling spiral way.How can I do this?Please advice
— Khairul, Wed, 23 Jan 2008 20:56:18 -0500 (EST)
I'm so glad to know it wasn't my lack of ability to figure this out...it really IS complicated. Sadly though, the problem no longer exists in Tulsa, OK. The only Charmin we can get now is the new Ultra Soft or Ultra Strong. Both are HIGHLY inferior to the previous choices. For the first time since 1960, I am switching brands. "Angel Soft" brand is a nice option. But I miss the old Charmin.
— Barbara Holman, Sun, 10 Feb 2008 14:57:39 -0500 (EST)