2013 AMC 8 Problem 25
Attempt Problem 25 of the 2013 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2013 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
25.
A ball with diameter inches starts at point to roll along the track shown. The track is comprised of 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 in inches the center of the ball travels over the course from to ?
Answer: A
Small hint:
The ball center follows offset semicircular arcs.
Big hint:
Subtract the ball radius from outside arcs and add it to the inside arc.
Video solution:
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Written solution:
The ball has radius . As the ball rolls, its center follows semicircular arcs parallel to the track. On the first and third arcs the center path is inside the track arcs, so those radii are and . On the middle arc the center path is outside the track arc, so that radius is .
The total distance traveled by the center is the sum of the three semicircle lengths:
Thus, A is the correct answer.
Problem 25 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
- 1994 AMC 8
- 1995 AMC 8
- 1996 AMC 8
- 1997 AMC 8
- 1998 AMC 8
- 1999 AMC 8
- 2000 AMC 8
- 2001 AMC 8
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- 2006 AMC 8
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