A circle is a curve which maintains same distance from a fixed point called center.
The perimeter of a circle is the length of the curve and area of a circle is portion of a plane bounded by the curve.
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?
$\textbf{(A)}\ 238\pi \qquad \textbf{(B)}\ 240\pi \qquad \textbf{(C)}\ 260\pi \qquad \textbf{(D)}\ 280\pi \qquad \textbf{(E)}\ 500\pi$
AMC 8 2013 Problem 25
Geometry : Perimeter of a circle
7 out of 10
Mathematical Circles.
First I want to give you the formula required.
You can clearly notice that we have to find the perimeters of all of the semicircles
The perimeter of a circle of radius $r$ unit can be obtained by the formula $2\pi r$. Then can you find perimeter of the semicircles ?!!!
So using the formula, the perimeters of
Semicircle 1 =$\frac{2\pi\times 100}{2}$ inches.
Semicircle 2 =$\frac{2\pi\times 60}{2}$ inches.
Semicircle 3 =$\frac{2\pi\times 80}{2}$ inches.
So the total path covered by the ball is
$\pi(100+60+80)=240\pi$ inches.
Is it the final answer??? Or have we ignored something ?
OK !!! please notice that they have asked for the distance covered by the center of the ball.
And the ball is of radius \(2\) inches.
So for the \(1^{st}\) and \(3^{rd}\) semicircle : The center will roll along a semicircular path of radius \(R_1-2\) and \(R_3-2\).
See this image :
And for the \(2^{nd}\) semicircle : The center will roll along a semicircular path of radius \(R_2+2\).
See the image below :
So the length of the path covered by the center of the ball is
\([\pi(100-2)+\pi(60+2)+\pi(80-2)] \quad \text{inches} \\=\pi(98+62+78) \quad \text{inches}\\=238\pi \quad \text{inches}\).
AMC - AIME Boot Camp... for brilliant students.
Use our exclusive one-on-one plus group class system to prepare for Math Olympiad
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]