Difference between revisions of "2013 AMC 8 Problems/Problem 24"
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==Problem== | ==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 <math>R_1 = 100</math> inches, <math>R_2 = 60</math> inches, and <math>R_3 = 80</math> 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? | ||
| + | <asy> | ||
| + | size(8cm); | ||
| + | draw((0,0)--(480,0),linetype("3 4")); | ||
| + | filldraw(circle((8,0),8),black); | ||
| + | draw((0,0)..(100,-100)..(200,0)); | ||
| + | draw((200,0)..(260,60)..(320,0)); | ||
| + | draw((320,0)..(400,-80)..(480,0)); | ||
| + | draw((100,0)--(150,-50sqrt(3)),Arrow(size=4)); | ||
| + | draw((260,0)--(290,30sqrt(3)),Arrow(size=4)); | ||
| + | draw((400,0)--(440,-40sqrt(3)),Arrow(size=4)); | ||
| + | </asy> | ||
| + | |||
| + | <math>\textbf{(A)}\ 238\pi \qquad \textbf{(B)}\ 240\pi \qquad \textbf{(C)}\ 260\pi \qquad \textbf{(D)}\ 280\pi \qquad \textbf{(E)}\ 500\pi</math> | ||
==Solution== | ==Solution== | ||
Revision as of 09:42, 27 November 2013
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?
![[asy] size(8cm); draw((0,0)--(480,0),linetype("3 4")); filldraw(circle((8,0),8),black); draw((0,0)..(100,-100)..(200,0)); draw((200,0)..(260,60)..(320,0)); draw((320,0)..(400,-80)..(480,0)); draw((100,0)--(150,-50sqrt(3)),Arrow(size=4)); draw((260,0)--(290,30sqrt(3)),Arrow(size=4)); draw((400,0)--(440,-40sqrt(3)),Arrow(size=4)); [/asy]](http://latex.artofproblemsolving.com/4/9/a/49a0d39a0081dee41766dd79c74d1a46a6528307.png)
Solution
See Also
| 2013 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 23 |
Followed by Problem 25 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
