Question

Rachel 和 Robert 在圆形跑道上跑步。Rachel 逆时针跑，每 $90$ 秒跑完一圈；Robert 顺时针跑，每 $80$ 秒跑完一圈。两人同时从起点线出发。在他们开始跑步后 $10$ 分钟到 $11$ 分钟之间的某个随机时刻，一名站在跑道内部的摄影师拍下一张照片，照片显示以起点线为中心的四分之一圈跑道。两人都出现在照片中的概率是多少？

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?

$\dfrac{1}{16}$

$\dfrac18$

$\dfrac{3}{16}$

$\dfrac14$

$\dfrac{5}{16}$

答案：C

知识点：几何概率路程、速度与时间

难度评级：1920

解答：

照片覆盖起点两侧各 $\dfrac18$ 圈。 $600$ 秒后，Rachel 距离起点线还差 $30$ 秒；她跑完 $\dfrac14$ 圈需 $22.5$ 秒，所以她在第 $10$ 分钟的 $30-11.25=18.75$ 秒到 $30+11.25=41.25$ 秒之间入镜。

$600$ 秒后，Robert 距离起点线还差 $40$ 秒；他跑完 $\dfrac14$ 圈需 $20$ 秒，所以他在该分钟的 $30$ 秒到 $50$ 秒之间入镜。

两人同时入镜的时间为 $30$ 秒到 $41.25$ 秒，长度为 $11.25$ 秒，占 $60$ 秒的比例为 $\dfrac{11.25}{60}=\dfrac{3}{16}$ 。

所以正确答案是 C。

The picture spans $\dfrac18$ lap on each side of the start. After $600$ seconds Rachel is $30$ seconds short of the line; running $\dfrac14$ lap in $22.5$ seconds, she is in view between $30-11.25=18.75$ and $30+11.25=41.25$ seconds of the $10$ th minute.

After $600$ seconds Robert is $40$ seconds from the line; running $\dfrac14$ lap in $20$ seconds, he is in view between $30$ and $50$ seconds.

Both appear between $30$ and $41.25$ seconds, a window of length $11.25$ out of $60,$ so the probability is $\dfrac{11.25}{60}=\dfrac{3}{16}.$

Thus, the correct answer is C.

2009 AMC 10B 第 23 题

答案：C

难度评级：1920

解答：

其他年份的第 23 题