2011 AMC 10B Problem 12

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}

Answer: A

Concepts:circumference distance rate and time

Difficulty rating: 1370

Small hint:

The straight portions cancel when comparing outside and inside paths

Big hint:

Only the two semicircular ends add extra distance

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

\frac{12\pi}{36}=\dfrac\pi3

meters per second.

Thus, A is the correct answer.