October 2007: On the right track (solution)
The right-hand wheels travel 2π" farther than the left-hand wheels.
The two pairs of wheels travel the same distance along the straight tracks, so we may as well ignore them. We can take the curved tracks and reassemble them to form a circle. The circumference of the inside rail is 2πr and the circumference of the outside rail is 2π(r+1), so the difference is 2π". (If the track is not convex, then each right-hand curve cancels a left-hand curve, so we can toss them both out, leaving a single circle.)
Math fact: This applies much more generally. If a vehicle drives counterclockwise along any curvy route so that it loops back to the starting position without ever crossing its tracks (think of a Grand Prix race track), then the right wheels travel 2πb farther than the left wheels (where b is the distance between the wheels).
Correct solutions: Ritoban Basu Thakur
The two pairs of wheels travel the same distance along the straight tracks, so we may as well ignore them. We can take the curved tracks and reassemble them to form a circle. The circumference of the inside rail is 2πr and the circumference of the outside rail is 2π(r+1), so the difference is 2π". (If the track is not convex, then each right-hand curve cancels a left-hand curve, so we can toss them both out, leaving a single circle.)
Math fact: This applies much more generally. If a vehicle drives counterclockwise along any curvy route so that it loops back to the starting position without ever crossing its tracks (think of a Grand Prix race track), then the right wheels travel 2πb farther than the left wheels (where b is the distance between the wheels).
Correct solutions: Ritoban Basu Thakur
posted by Dave at 10:49 AM
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