2013 AMC 8 第 25 题
先试着解答 2013 AMC 8 第 25 题，然后核对你的答案与视频与精心整理的解答，解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 2013 AMC 8 解答，或核对答案。
所有题目均经美国数学协会（MAA）官方合法授权使用。
25.
一个直径为 英寸的球从点 出发沿图示轨道滚动。轨道由 段半圆弧组成，半径分别为 英寸、 英寸、 英寸。球始终与轨道接触且不打滑。从 到 的过程中，球心行进的距离是多少英寸？
A ball with diameter inches starts at point to roll along the track shown. The track is comprised of semicircular arcs whose radii are inches, inches, and 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 to ?
小提示：
球心沿着偏移后的半圆弧运动
The ball center follows offset semicircular arcs.
大提示：
对外侧圆弧减去球半径，对内侧圆弧加上球半径
Subtract the ball radius from outside arcs and add it to the inside arc.
视频讲解：
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文字解答：
球的半径为 。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 和 。中间圆弧中球心路径在轨道圆弧外侧，所以半径为 。
球心行进的总距离是三段半圆弧长度之和：
所以正确答案是 A。
The ball has radius . 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 and . On the middle arc the center path is outside the track arc, so that radius is .
The total distance traveled by the center is the sum of the three semicircle lengths:
Thus, A is the correct answer.
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