AMC10 2009 B

AMC10 2009 B · Q23

AMC10 2009 B · Q23. It mainly tests Rates (speed), Probability (basic).

Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the start line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?

Rachel 和 Robert 在一个圆形跑道上跑步。Rachel 逆时针跑，每 90 秒完成一圈，Robert 顺时针跑，每 80 秒完成一圈。他们同时从起点开始。在他们开始跑后 10 分钟到 11 分钟之间的某个随机时刻，站在跑道内侧的摄影师拍了一张照片，照片显示以起点线为中心的一刻跑道。Rachel 和 Robert 同时出现在照片中的概率是多少？

(A) $\frac{1}{16}$ $\frac{1}{16}$

(B) $\frac{1}{8}$ $\frac{1}{8}$

(D) $\frac{1}{4}$ $\frac{1}{4}$

(E) $\frac{5}{16}$ $\frac{5}{16}$

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): After 10 min. $=600$ sec., Rachel will have completed 6 laps and be 30 seconds from the finish line. Because Rachel runs one-fourth of a lap in 22.5 seconds, she will be in the picture taking region between $$ 30-\frac{22.5}{2}=18.75 \quad \text{and} \quad 30+\frac{22.5}{2}=41.25 $$ seconds of the 10th minute. After 10 minutes Robert will have completed 7 laps and will be 40 seconds from the starting line. Because Robert runs one-fourth of a lap in 20 seconds, he will be in the picture taking region between 30 and 50 seconds of the 10th minute. Hence both Rachel and Robert will be in the picture if it is taken between 30 and 41.25 seconds of the 10th minute. The probability that the picture is snapped during this time is $$ \frac{41.25-30}{60}=\frac{3}{16}. $$

答案（C）：10 分钟（$=600$ 秒）后，Rachel 将完成 6 圈，并距离终点线还有 30 秒。由于 Rachel 跑四分之一圈需要 22.5 秒，所以在第 10 分钟内，她出现在拍照区域的时间在 $$ 30-\frac{22.5}{2}=18.75 \quad \text{和} \quad 30+\frac{22.5}{2}=41.25 $$ 秒之间。10 分钟后，Robert 将完成 7 圈，并距离起点线还有 40 秒。由于 Robert 跑四分之一圈需要 20 秒，所以在第 10 分钟内，他出现在拍照区域的时间在 30 到 50 秒之间。因此，如果照片在第 10 分钟的 30 到 41.25 秒之间拍摄，Rachel 和 Robert 都会出现在照片中。照片在这段时间内拍下的概率为 $$ \frac{41.25-30}{60}=\frac{3}{16}. $$

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