AMC 8 2014 Problem 25

AMC8 2014 年第 25 题

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A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch? Note: 1 mile = 5280 feet

一段长为 1 英里，宽 40 英尺的高速公路被关闭了。Robert 沿着如下图中由一系列半圆组成的路径骑着自行车。若他骑车的速度是 5 英里每小时，那么走过这段 1 英里的高速公路需要多少小时？注意：1 英里＝5280 英尺

A. $\displaystyle \frac{\pi}{11}$
B. $\displaystyle \frac{\pi}{10}$
C. $\displaystyle \frac{\pi}{5}$
D. $\displaystyle \frac{2\pi}{5}$
E. $\displaystyle \frac{2\pi}{3}$

正确答案 · Correct Answer

解析 · Solution

解法1

常规解法 · 数半圆每个半圆直径等于路宽 40 英尺，半径 20。一英里 =5280 英尺，含 5280/40=132 个半圆。每个弧长

\pi\cdot20=20\pi

英尺，总长

132×20\pi=2640\pi

英尺

=\dfrac{2640\pi}{5280}=\dfrac{\pi}{2}

英里。时间

\dfrac{\pi/2}{5}=\dfrac{\pi}{10}

小时。

Solution 1

Elementary · count the semicircles Each semicircle has diameter 40 ft (the road width), radius 20. A mile is 5280 ft, giving 5280/40=132 semicircles. Each arc is

\pi\cdot20=20\pi

ft, so the total is

132\cdot20\pi=2640\pi

=\dfrac{2640\pi}{5280}=\dfrac{\pi}{2}

miles. The time is

\dfrac{\pi/2}{5}=\dfrac{\pi}{10}

hours.

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