Math Stumpers

We need a function which will follow the curve of the paper wrapped around the roll. Then we can simply calculate its arc length to find out how long the paper is. Since the paper follows an expanding spiral, it would be difficult to write the function in Cartesian coordinates. Instead, we'll use polar coordinates, which means that the distance from the origin, r, is a function of the angle theta as measured from the positive x axis. Finding the right function isn't easy, but we can start with a simple function and work from there. We're going to use radians instead of degrees because when we want to determine arc length, radian measure makes our job easier. If we have the function

then as the angle theta increases, the path of the function will simply trace a circle, not the spiral we need. We want the function to start at r = v when theta is zero, but by the time theta increases to 2, our function should have the value r = v + 0.001, because the paper thickness is 1 mm. It looks as though we need r to be the sum of v and a function of theta. After a few false starts, we try

and examine it to see whether it will work for us. This function is a spiral which will move out 1 mm from the origin during the course of one cycle, which fits nicely with how the roll of paper looks. Of course the curve this function traces has no width, but we can vary r so that it exactly traces the path of the paper. Now that we have the function we want, we will find its arc length.

Start slowly: if you have a constant radius and rotate it through 2 radians, you would simply trace a circle. We know that a circle's arc length is the same as its circumference, 2r. If we have half of a circle (which contains radians), the arc length will be half the circumference of a full circle.

If we have any part of a circle which contains theta radians, then its arc length is

because a fractional part of any circle produces a fractional part of the circumference of that circle.

If we have a tiny part of a circle with an infinitesimal part of a radian, then the arc length will be infinitesimal as well, and it will be of the form

This equality is a way of saying that an infinitesimal arc length is equal to the radius times an infinitesimal radian measure. Happily, even if r is not a constant, it will behave like a constant over a very small change in theta (see note at end of solution). In general, we can write

Now, to find the length of the paper, we need to know which values of theta in our problem are appropriate for the integration of the equality above. What are those values? They are the values which satisfy the initial state, r = v, and the final state, r = w, and all values between those bounds. We designed the function so that it would assume v when theta is zero, and we now solve for the final value of theta, the one such that

Next we find the arc length, s, by adding up all of the infinitesimal bits of arc length. We integrate

and after evaluation, this simple expression is the length of paper on the roll. In the case where v = 3 cm and w = 13 cm, this length becomes

 

Note: the claim that r behaves like a constant for small changes in theta was not proven here. The claim is true for vectors of the form

where

is a continuous function whose domain is a connected subset of the real numbers (and is also true for a broader class of functions, but not for every function). The function we used in this solution satisfies this criteria.

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