If the arcs of the same length in two circles subtend angles \(65^\circ\) and \(110^\circ\) at the center, find the ratio of their radii.
Solution:
We know:
\[ s=r\theta \]
Since the arc lengths are equal,
\[ r_1\theta_1=r_2\theta_2 \]
Therefore,
\[ \frac{r_1}{r_2}=\frac{\theta_2}{\theta_1} \]
Given:
\[ \theta_1=65^\circ \]
\[ \theta_2=110^\circ \]
So,
\[ \frac{r_1}{r_2}=\frac{110}{65} \]
\[ \frac{r_1}{r_2}=\frac{22}{13} \]
Therefore, the ratio of the radii is:
\[ \boxed{22:13} \]