AMC 8 2013 Problem 25
AMC8 2013 年第 25 题
圆★★★★☆
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 the center of the ball travels over the course from A to B?
直径为 4 英寸的球从 A 开始沿着图示轨道滚动。轨道由 3 段半圆弧组成，半径分别为 R1=100 英寸，R2=60 英寸，R3=80 英寸。球全程都和轨道紧密接触，并且不会滑动。那么当球从 A 滚到 B，球心所经过的路程是多少英寸？
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- B.
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正确答案 · Correct Answer
A
解析 · Solution
解法1
常规解法 · 逐段算圆心半径
球半径 2，球心也走半圆，但半径随轨道凹凸变化：向背离球的方向弯时半径减 2，向球这侧弯时加 2。三段依次为 100-2=98、60+2=62、80-2=78，球心总路程 。
巧解 · 先求弧长再净调整
三段半圆弧长之和 。凹凸交替使球心半径相对轨道净偏移 -2+2-2=-2，故球心路程 。
Solution 1
Elementary · each arc's center radius
The ball has radius 2, so its center also traces semicircles, with radius shifted by 2: smaller by 2 where the track curves away from the ball, larger by 2 where it curves toward it. The radii are 100-2=98, 60+2=62, 80-2=78, so the center travels .
Slick · arc lengths, then net adjustment
The three arcs total . The alternating curvature shifts the center's radius by -2+2-2=-2, so the center path is .
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