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。两人同时从起点出发。开始跑步后 $10$ 分钟到 $11$ 分钟之间的某个随机时刻，一位站在跑道内的摄影师拍了一张照片，照片显示以起跑线为中心的四分之一圈跑道。两人都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?

$\dfrac{1}{16}$

$\dfrac{1}{8}$

$\dfrac{3}{16}$

$\dfrac{1}{4}$

$\dfrac{5}{16}$

答案：C

知识点：几何概率路程、速度与时间

难度评级：1930

解答：

照片覆盖距离起点不超过 $\dfrac{1}{8}$ 圈的弧。在 $600$ 秒时，Rachel 已跑 $6\tfrac{2}{3}$ 圈，距离起点还差 $30$ 秒；四分之一圈对应 $22.5$ 秒，所以她在 $30 - 11.25 = 18.75$ 秒到 $30 + 11.25 = 41.25$ 秒之间入镜，这是第 $10$ 分钟内的时间。

在 $600$ 秒时，Robert 距离起跑线 $40$ 秒；四分之一圈需要 $20$ 秒，所以他在 $30$ 到 $50$ 秒之间入镜。两人都入镜的时间是 $30$ 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 covers the arc within $\dfrac{1}{8}$ lap of the start on each side. At $600$ s Rachel has run $6\tfrac{2}{3}$ laps, $30$ s short of the line; a quarter lap takes her $22.5$ s, so she is in view between $30 - 11.25 = 18.75$ s and $30 + 11.25 = 41.25$ s of the $10$ th minute.

At $600$ s Robert is $40$ s from the line; a quarter lap takes $20$ s, so he is in view between $30$ and $50$ s. Both appear between $30$ and $41.25$ s, a window of $11.25$ s out of $60,$ giving probability $\dfrac{11.25}{60} = \dfrac{3}{16}.$

Thus, the correct answer is C.

2009 AMC 12B 第 18 题

答案：C

难度评级：1930

解答：

其他年份的第 18 题