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Difference between revisions of "2013 AMC 8 Problems/Problem 24"

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==Problem==
 
==Problem==
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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?
 
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<asy>
 
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size(8cm);
 
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draw((0,0)--(480,0),linetype("3 4"));
 
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filldraw(circle((8,0),8),black);
 
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draw((0,0)..(100,-100)..(200,0));
 
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draw((200,0)..(260,60)..(320,0));
 
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draw((320,0)..(400,-80)..(480,0));
 
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draw((100,0)--(150,-50sqrt(3)),Arrow(size=4));
 
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draw((260,0)--(290,30sqrt(3)),Arrow(size=4));
 
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draw((400,0)--(440,-40sqrt(3)),Arrow(size=4));
 
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</asy>
 
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<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 08:46, 27 November 2013

Problem

Solution

The radius of the ball is 2 inches. If you think about the ball rolling or draw a path for the ball, you see that in A and C it loses $2\pi*2/2=2\pi$ inches, and it gains $2\pi$ inches on B. So, the departure from the length of the track means that the answer is $\frac{200+120+160}{2}*\pi+(-2-2+2)*\pi=240\pi-2\pi=\boxed{\textbf{(A)}\ 238\pi}$.

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

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