2006 AMC 10A Problem 15

Question

Odell and Kershaw run for

30

minutes on a circular track. Odell runs clockwise at

250

m/min and uses the inner lane with a radius of

50

meters. Kershaw runs counterclockwise at

300

m/min and uses the outer lane with a radius of

60

meters, starting on the same radial line as Odell. How many times after the start do they pass each other?

29

42

45

47

50

Compute each runner’s time for one lap; they turn out equal

Running in opposite directions, they meet twice per lap; count how many laps fit in

30

minutes

Odell’s lap is

2\pi(50) = 100\pi

m at

250

m/min, taking

\frac{100\pi}{250} = 0.4\pi

min. Kershaw’s lap is

2\pi(60) = 120\pi

m at

300

m/min, also

\frac{120\pi}{300} = 0.4\pi

min.

Their periods are equal. Running in opposite directions, they meet at times

t = \frac{k}{2}(0.4\pi)

for

k = 1, 2, \ldots

Requiring

t \le 30

gives

k \le \frac{60}{0.4\pi} = \frac{150}{\pi} \approx 47.7,

so they pass

47

times.

Thus, the correct answer is D.

Problem 15 in other years