Question

Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?

$\dfrac{\pi}{3}$

$\dfrac{2\pi}{3}$

$\pi$

$\dfrac{4\pi}{3}$

$\dfrac{5\pi}{3}$

Solution:

Let the inner semicircle radius be $r$ . The straight parts of the inside and outside paths have the same total length, so only the semicircular ends change the distance.

The two inner semicircles have total length $2\pi r$ , while the two outer semicircles have total length $2\pi(r+6)$ . The outside path is therefore $12\pi$ meters longer.

Keiko takes $36$ more seconds to walk $12\pi$ more meters, so her speed is $12\pi/36=\dfrac\pi3$ meters per second.

Thus, A is the correct answer.

2011 AMC 10B Problem 12

Solution:

Problem 12 in Other Years