2006 AMC 10A Problem 15

Problem 15 of 25IntermediateAlgebra Geometry

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?

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Solution

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.