MathDB

25

Part of 2013 AMC 8

Problems(1)

2013/8/25/Rolling on a semicircle

Source:

11/27/2013
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 R1=100R_1 = 100 inches, R2=60R_2 = 60 inches, and R3=80R_3 = 80 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("R1R_1",(100,0)--(150,-50sqrt(3)), W, fontsize(10)); label("R2R_2",(260,0)--(290,30sqrt(3)), W, fontsize(10)); label("R3R_3",(400,0)--(440,-40sqrt(3)), W, fontsize(10)); filldraw(circle((8,0),8),black); label("AA",(0,0),W,fontsize(10));[/asy]
<spanclass=′latex−bold′>(A)</span> 238π<spanclass=′latex−bold′>(B)</span> 240π<spanclass=′latex−bold′>(C)</span> 260π<spanclass=′latex−bold′>(D)</span> 280π<spanclass=′latex−bold′>(E)</span> 500π<span class='latex-bold'>(A)</span>\ 238\pi \qquad <span class='latex-bold'>(B)</span>\ 240\pi \qquad <span class='latex-bold'>(C)</span>\ 260\pi \qquad <span class='latex-bold'>(D)</span>\ 280\pi \qquad <span class='latex-bold'>(E)</span>\ 500\pi
AMCAIMEAMC 8AMC 10