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Try this beautiful problem from Geometry based on a rolling ball on a semicircular track.

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 the center of the ball travels over the course from A to B?

- \( 235 \pi\)
- \( 238\pi\)
- \( 240 \pi\)

Geometry

circumference of a semicircle

Circle

But try the problem first...

Answer:\( 238 \pi\)

Source

Suggested Reading

AMC-8, 2013 problem 25

Pre College Mathematics

First hint

Find the circumference of semicircle....

Can you now finish the problem ..........

Second Hint

Find the total distance by the ball....

can you finish the problem........

Final Step

The radius of the ball is 2 inches. If you think about the ball rolling or draw a path for the ball (see figure below), you see that in A and C it loses \(2\pi \times \frac{2}{2}=2\pi\) inches each, and it gains \(2\pi\) inches on B .

So, the departure from the length of the track means that the answer is

\(\frac{200+120+160}{2} \times \pi\) + (-2-2+2) \(\times \pi\)=240\(\pi\) -2\(\pi\)=238\(\pi\)

- https://www.cheenta.com/area-of-a-square-amc-8-2015-problem-25/
- https://www.youtube.com/watch?v=W9XdZd8zXPA

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