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jmc

geometry junior

Problem

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 inches, inches, and 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?
problem
(A)
(B)
(C)
(D)
Solution
The total length of all of the arcs is . Since we want the path from the center, the actual distance will be subtracted by 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 Therefore, the answer is .
Final answer
A