The radius of a circle is \(30\) cm. Find the length of an arc of this circle, if the length of the chord of the arc is \(30\) cm.
Solution:
Radius of the circle:
\[ r=30 \text{ cm} \]
Chord length:
\[ c=30 \text{ cm} \]
Using chord formula:
\[ c=2r\sin\frac{\theta}{2} \]
Substituting the values:
\[ 30=2(30)\sin\frac{\theta}{2} \]
\[ 30=60\sin\frac{\theta}{2} \]
\[ \sin\frac{\theta}{2}=\frac{1}{2} \]
\[ \frac{\theta}{2}=\frac{\pi}{6} \]
\[ \theta=\frac{\pi}{3} \]
Now, arc length:
\[ s=r\theta \]
\[ s=30\times\frac{\pi}{3} \]
\[ s=10\pi \text{ cm} \]
Therefore, the length of the arc is:
\[ \boxed{10\pi \text{ cm}} \]