In the diagram, $A C=C B=10 \mathrm{~m}$, where $A C$ and $C B$ are each the diameter of the small equal semi-circles. The diameter of the larger semi-circle is $A B$. In travelling from $A$ to $B$, it is possible to take one of two paths. One path goes along the semi-circular arc from $A$ to $B$. A second path goes along the semi-circular arcs from $A$ to $C$ and then along the semi-circular arc from $C$ to $B$. The difference in the lengths of these two paths is
(A) $12 \pi$
(B) $6 \pi$
(C) $3 \pi$
(D) $2 \pi$
(E) 0

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-101.jpg?height=318&width=439&top_left_y=871&top_left_x=1323)
In the diagram, $A C=C B=10 \mathrm{~m}$, where $A C$ and $C B$ are each the diameter of the small equal semi-circles. The diameter of the larger semi-circle is $A B$. In travelling from $A$ to $B$, it is possible to take one of two paths. One path goes along the semi-circular arc from $A$ to $B$. A second path goes along the semi-circular arcs from $A$ to $C$ and then along the semi-circular arc from $C$ to $B$. The difference in the lengths

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-453.jpg?height=366&width=520&top_left_y=291&top_left_x=1388)
of these two paths is
(A) $12 \pi$
(B) $6 \pi$
(C) $3 \pi$
(D) $2 \pi$
(E) 0

## Solution

Consider the two calculations.

## Calculation 1

The distance travelled here would be one-half the circumference of the circle with radius 10.

This distance would be $\frac{1}{2}[2 \pi(10)]=10 \pi$.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-453.jpg?height=309&width=520&top_left_y=1103&top_left_x=1388)

## Calculation 2

The distance travelled would be the equivalent to the circumference of a circle with radius 5 . The distance would be $2 \pi(5)=10 \pi$.

Since these distances are equal, their difference would be $10 \pi-10 \pi=0$.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-453.jpg?height=250&width=542&top_left_y=1512&top_left_x=1388)

ANSWER: (E)