Question

Emily 看见一艘船以恒定速度沿河的一段直线航行。她以比船更快的匀速平行于河岸行走。从船尾走到船头时，她数了 $210$ 个等长步幅；反方向从船头走到船尾时，她数了 $42$ 个同样大小的步幅。用 Emily 的步幅作单位，这艘船的长度是多少？

Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts $42$ steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship?

$70$

$84$

$98$

$105$

$126$

答案：A

知识点：相对速度路程、速度与时间比与比例

难度评级：1370

解答：

设船长为 $x$ 个步幅。Emily 走 $210$ 步从船尾到船头时，船也向前移动了 $210 - x$ 个步幅。

反方向走 $42$ 步时，船移动了 $x - 42$ 个步幅。船速与 Emily 速度恒定，所以 $\dfrac{210}{210 - x} = \dfrac{42}{x - 42}.$ $210x - 210 \cdot 42 = 210 \cdot 42 - 42x$ $42 \cdot 6 x = 2 \cdot 42 \cdot 210$ $x = 70.$

所以正确答案是 A。

Let $x$ be the length of the ship. Then in the time that Emily moves $210$ steps, the ship moves $210 - x$ steps.

In the time that Emily moves $42$ steps, the ship moves $x - 42$ steps. Since the ship and Emily move at a constant rate $\dfrac{210}{210 - x} = \dfrac{42}{x - 42}.$ Cross-multiplying yields $210x - 210 \cdot 42 = 210 \cdot 42 - 42x$ $42 \cdot 6 x = 2 \cdot 42 \cdot 210$ $x = 70.$

Thus, A is the correct answer.

2021 AMC 10A Fall 第 11 题

答案：A

难度评级：1370

解答：

其他年份的第 11 题