Question

一个直径为 $4$ 英寸的球从点 $A$ 出发沿图示轨道滚动。轨道由 $3$ 段半圆弧组成，半径分别为 $R_1 = 100$ 英寸、 $R_2 = 60$ 英寸、 $R_3 = 80$ 英寸。球始终与轨道接触且不打滑。从 $A$ 到 $B$ 的过程中，球心行进的距离是多少英寸？

A ball with diameter $4$ inches starts at point $A$ to roll along the track shown. The track is comprised of $3$ semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance in inches the center of the ball travels over the course from $A$ to $B$ ?

$238 \pi$

$240 \pi$

$260 \pi$

$280 \pi$

$500 \pi$

答案：A

知识点：弧圆周长

难度评级：1930

视频讲解：

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文字解答：

球的半径为 $2$ 。滚动时，球心沿着与轨道平行的半圆弧运动。第一段和第三段圆弧中球心路径在轨道圆弧内侧，所以半径分别为 $100-2=98$ 和 $80-2=78$ 。中间圆弧中球心路径在轨道圆弧外侧，所以半径为 $60+2=62$ 。

球心行进的总距离是三段半圆弧长度之和： $(98+62+78)\pi=238\pi.$

所以正确答案是 A。

The ball has radius $2$ . As the ball rolls, its center follows semicircular arcs parallel to the track. On the first and third arcs the center path is inside the track arcs, so those radii are $100-2=98$ and $80-2=78$ . On the middle arc the center path is outside the track arc, so that radius is $60+2=62$ .

The total distance traveled by the center is the sum of the three semicircle lengths: $(98+62+78)\pi=238\pi.$

Thus, A is the correct answer.

2013 AMC 8 第 25 题

答案：A

难度评级：1930

视频讲解：

文字解答：

其他年份的第 25 题