Question

一条笔直的河宽 $264$ 米，水流以每分钟 $14$ 米的速度自西向东流。Melanie 和 Sherry 坐在河的南岸， Melanie 位于 Sherry 下游 $D$ 米处。相对于水，Melanie 的游泳速度为每分钟 $80$ 米，Sherry 的游泳速度为每分钟 $60$ 米。两人同时开始沿直线游向北岸上的同一点，该点到她们的起点距离相等。两人同时到达该点。求 $D$ 。

A straight river that is $264$ meters wide flows from west to east at a rate of $14$ meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of $D$ meters downstream from Sherry. Relative to the water, Melanie swims at $80$ meters per minute, and Sherry swims at $60$ meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find $D.$

答案

答案：550

知识点：路程、速度与时间向量平方差

难度评级：2390

解答：

把 Sherry 放在原点，Melanie 放在南岸的 $(D, 0)$ 。北岸上到两人等距的点是 $\left(\frac{D}{2}, 264\right)$ 。若两人都在时间 $t$ ，后到达，则每位游泳者相对于水的速度等于她的对地速度减去水流速度 $(14, 0)$ ，所以 $\begin{aligned} &\left(\frac{D}{2t} - 14\right)^2 \\ &\quad {}+ \left(\frac{264}{t}\right)^2 = 60^2, \end{aligned}$ $\begin{aligned} &\left(-\frac{D}{2t} - 14\right)^2 \\ &\quad {}+ \left(\frac{264}{t}\right)^2 = 80^2. \end{aligned}$

令 $u = \frac{D}{2t}$ ：并相减，得到 $(u + 14)^2 - (u - 14)^2$ $= 56u$ $= 6400 - 3600$ $= 2800$ ，所以 $u = 50$ 。代回可得 $(50 - 14)^2 + \left(\frac{264}{t}\right)^2 = 3600$ ，因此 $\frac{264}{t} = 48$ ， $t = \frac{11}{2}$ 。

因此 $D = 2ut = 100t = 550$ 。

Put Sherry at the origin and Melanie at $(D, 0)$ on the south bank. A point on the north bank equidistant from both is $\left(\frac{D}{2}, 264\right).$ If both arrive at time $t,$ then each swimmer's velocity relative to the water is her ground velocity minus the current $(14, 0),$ so $\begin{aligned} &\left(\frac{D}{2t} - 14\right)^2 \\ &\quad {}+ \left(\frac{264}{t}\right)^2 = 60^2, \end{aligned}$ $\begin{aligned} &\left(-\frac{D}{2t} - 14\right)^2 \\ &\quad {}+ \left(\frac{264}{t}\right)^2 = 80^2. \end{aligned}$

Subtracting, with $u = \frac{D}{2t}:$ $(u + 14)^2 - (u - 14)^2$ $= 56u$ $= 6400 - 3600$ $= 2800,$ so $u = 50.$ Substituting back, $(50 - 14)^2 + \left(\frac{264}{t}\right)^2 = 3600$ gives $\frac{264}{t} = 48,$ so $t = \frac{11}{2}.$

Therefore $D = 2ut = 100t = 550.$

2022 AIME I 第 5 题

答案：550

难度评级：2390

解答：

其他年份的第 5 题