2011 AMC 10B Problem 12

We are not told how long the track is, so look only at the part of the path that changes between the two edges.

1. The straight parts are the same. The inside and outside edges run along the same two straight sides, so they have the same straight length. That length will cancel.
2. The two round ends make one full circle. Two half circles at the ends together make one full circle.
3. Say the inside circle has radius $r$. The track is $6$ meters wide, so the outside circle has radius $r + 6$.
4. The inside circle is $2\pi r$ long and the outside circle is $2\pi(r + 6) = 2\pi r + 12\pi$ long.
5. So the outside edge is exactly $12\pi$ meters longer than the inside edge, whatever $r$ is.
6. Find the speed. Those extra $12\pi$ meters take her $36$ extra seconds. Speed is distance divided by time:
$$\frac{12\pi}{36} = \frac{\pi}{3} \text{ meters per second}.$$

The answer is $\boxed{\textbf{(A)}\ \frac{\pi}{3}}$.

A circle's length is $2\pi$ times its radius, so making the radius $6$ bigger always adds $2\pi \times 6 = 12\pi$, no matter how big the circle was. When a problem leaves out a measurement you think you need, write both distances with a letter for it and subtract; the missing measurement usually cancels.