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?

$29$

$42$

$45$

$47$

$50$

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.

2006 AMC 10A Problem 15

Solution:

Problem 15 in Other Years