2013/8/25/Rolling on a semicircle
Source:
November 27, 2013
AMCAIMEAMC 8AMC 10
Problem Statement
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));
label("",(100,0)--(150,-50sqrt(3)), W, fontsize(10));
label("",(260,0)--(290,30sqrt(3)), W, fontsize(10));
label("",(400,0)--(440,-40sqrt(3)), W, fontsize(10));
filldraw(circle((8,0),8),black);
label("",(0,0),W,fontsize(10));[/asy]