# The rate change of the radius of a coil.

Suppose I have a tube of radius $r_0$ that I want to wrap a sheet of length $l$ and thickness $\Delta x$.

Assuming the radius changes only when the paper overlaps the where the previous section overlapped, how does the radius change?

If you are having trouble visualizing this, consider a roll of toilet paper. How does the radius of the roll change as I remove (or add) a fixed length of paper?

The easiest way would be to invert the relation. Since we know that the radius of the roll $r$ increases with respect to the sheet length $l$, we can find $l$ in terms of $r$ first. That's easy because if we look along the axis, the total area taken up by the paper is approximately $π(r^2-{r_0}^2)$, so the total length is about $\frac{π(r^2-{r_0}^2)}{Δx}$. This is a first-order approximation of course, since we assume that $Δx$ is small enough. So the answer would be $r \approx \sqrt{\frac{lΔx}{π}+r_0^2}$