AMC8 2013

AMC8 2013 · Q25

AMC8 2013 · Q25. It mainly tests Circle theorems, Geometry misc.

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?

一个直径为 4 英寸的球从点 A 开始沿所示轨道滚动。轨道由 3 个半圆弧组成，半径分别为 $R_1 = 100$ 英寸、$R_2 = 60$ 英寸和 $R_3 = 80$ 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) 238\pi 238\pi

(B) 240\pi 240\pi

(D) 280\pi 280\pi

(E) 500\pi 500\pi

Answer

Correct choice: (A)

正确答案：（A）

Solution

The total length of all of the arcs is $100\pi +80\pi +60\pi=240\pi$. Since we want the path from the center, the actual distance will be subtracted by $2\pi$ because it's already half the circumference through semicircle A, which needs to go half the circumference extra through semicircle B, and it's already half the circumference through semicircle C, and the circumference is $4\pi$ Therefore, the answer is $240\pi-2\pi=\boxed{\textbf{(A)}\ 238\pi}$. Video Solution: https://www.youtube.com/watch?v=zZGuBFyiQrk by WhyMath

所有弧的总长度是 $100\pi +80\pi +60\pi=240\pi$。由于要计算球心的路径，需要减去 $2\pi$，因为它已经通过半圆 A 的半周长，需要额外通过半圆 B 的半周长，已经通过半圆 C 的半周长，周长为 $4\pi$。因此，答案是 $240\pi-2\pi=\boxed{\textbf{(A)}\ 238\pi}$。

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