Problem 25 · 2013 AMC 8
Hard
Geometry & Measurement
rolling-ball-patharc-length
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 R1 = 100 inches, R2 = 60 inches, and R3 = 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?
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Answer: A — 238π.
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Hint 1 of 2
The ball touches the track but its center always sits one ball-radius (2 in) away. So the center traces its own set of semicircles, each concentric with the track's — the only question is whether the center is on the near or far side of each arc.
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Hint 2 of 2
On an arc that curves away from the center of the ball's path (ball rolls along the outside), the center's radius is R − 2; where the track curves the other way (ball rides the inside of the bend), it's R + 2. Match each arc to the picture, then a semicircle's length is just πR.
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Approach: shift each arc's radius by the ball's radius, then add the half-circumferences
- The ball's diameter is 4, so its radius is 2 — the center always floats 2 inches off the track.
- Read each bend from the figure: arc 1 and arc 3 are humps the ball rolls over the outside of, so the center's radius shrinks by 2 (100 → 98, 80 → 78); arc 2 is the dip where the ball hugs the inside, so it grows by 2 (60 → 62).
- Each leg is a semicircle of length π·(center radius), so total = π(98 + 62 + 78) = 238π.
- Why the ±2 matters: ignoring the offset gives π(100+60+80) = 240π (choice B, the trap). The shifts nearly cancel but not quite — two shrinks and one grow leave a net −2π.
- You'll see this again: a rolling object's center traces a curve parallel to the track, offset by its radius — the same idea behind a coin rolling around a coin.
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