2011 AMC 12B Problem 8

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 width

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}

Answer: A

Concepts:circumference distance rate and time

Difficulty rating: 1330

Small hint:

The straight portions have the same length on both edges, so only the curved ends differ

Big hint:

The two semicircular ends form full circles whose radii differ by

6

Solution:

The straight sides are the same length for both paths, so the difference in length comes only from the two semicircular ends. If the inner radius is

r,

those ends combine into a full circle, and the extra length is

2\pi(r+6)-2\pi r=12\pi.

If her speed is

x

meters per second, then the extra time gives

36x=12\pi,

so

x=\dfrac{\pi}{3}.

Thus, the correct answer is A.