AMC8 2014

AMC8 2014 · Q25

AMC8 2014 · Q25. It mainly tests Rates (speed), Circle theorems.

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

一条直直的一英里长、高度为40英尺的公路被封闭。罗伯特骑自行车沿着如图所示由半圆组成的路径前进。如果他的骑行速度为每小时5英里，他需要多少小时才能骑完这一英里路程？注：1 英里 = 5280 英尺

(A) \frac{\pi}{11} \frac{\pi}{11}

(B) \frac{\pi}{10} \frac{\pi}{10}

(D) \frac{2\pi}{5} \frac{2\pi}{5}

(E) \frac{2\pi}{3} \frac{2\pi}{3}

Answer

Correct choice: (B)

正确答案：（B）

Solution

There are two possible interpretations of the problem: that the road as a whole is $40$ feet wide, or that each lane is $40$ feet wide. Both interpretations will arrive at the same result. However, let us stick with the first interpretation for simplicity. Each lane must then be $20$ feet wide, so Robert must be riding his bike in semicircles with radius $20$ feet and diameter $40$ feet. Since the road is $5280$ feet long, over the whole mile, Robert rides $\frac{5280}{40} =132$ semicircles in total. Were the semicircles full circles, their circumference would be $2\pi\cdot 20=40\pi$ feet; as it is, the circumference of each is half that, or $20\pi$ feet. Therefore, over the stretch of highway, Robert rides a total of $132\cdot 20\pi =2640\pi$ feet, which is equivalent to $\frac{\pi}{2}$ miles. Robert rides at 5 miles per hour, so divide the $\frac{\pi}{2}$ miles by $5$ mph (because $t = \frac{d}{r}$ and time = distance/rate) to arrive at $\boxed{\textbf{(B) }\frac{\pi}{10}}$ hours. Edit: If you are confused about the lanes, watch the video below :)

这个问题有两种可能的理解：要么整个道路宽度为 $40$ 英尺，要么每条车道宽度为 $40$ 英尺。两种理解方式最终都会得出相同的结果。但为了简单起见，我们采用第一种理解。这样每条车道宽度为 $20$ 英尺，因此罗伯特必须骑行半径为 $20$ 英尺、直径为 $40$ 英尺的半圆。由于道路全长为 $5280$ 英尺，整个一英里内，罗伯特总共骑行了 $\frac{5280}{40} =132$ 个半圆。如果这些半圆是完整的圆圈，它们的周长应为 $2\pi \cdot 20=40\pi$ 英尺；而实际每个半圆周长是其一半，即 $20\pi$ 英尺。因此，在这段公路上，罗伯特共骑行了 $132 \cdot 20\pi =2640\pi$ 英尺，相当于 $\frac{\pi}{2}$ 英里。罗伯特骑行速度为每小时5英里，所以时间为距离除以速度，即将 $\frac{\pi}{2}$ 英里除以 $5$ 英里/小时，得出时间为 $\boxed{\textbf{(B) }\frac{\pi}{10}}$ 小时。编辑：如果你对车道宽度有疑问，可以观看下面的视频 :)

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