Question

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 $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ 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 $A$ to $B$ ?

$238 \pi$

$240 \pi$

$260 \pi$

$280 \pi$

$500 \pi$

Video solution:

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Written solution:

The ball has radius $2$ . 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 $100-2=98$ and $80-2=78$ . On the middle arc the center path is outside the track arc, so that radius is $60+2=62$ .

The total distance traveled by the center is the sum of the three semicircle lengths: $(98+62+78)\pi=238\pi.$

Thus, A is the correct answer.

2013 AMC 8 Problem 25

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Problem 25 in Other Years