The Radius of the Circle Whose Arc of Length \(15\pi\) cm Makes an Angle of \(\frac{3\pi}{4}\) Radian at the Centre

Question:

The radius of the circle whose arc of length \(15\pi\) cm makes an angle of \(\frac{3\pi}{4}\) radian at the centre is

(a) \(10\) cm

(b) \(20\) cm

(c) \(11 \frac{1}{4}\) cm

(d) \(22 \frac{1}{2}\) cm

Solution

We know the formula for arc length:

\[ l = r\theta \]

Given:

\[ l = 15\pi \text{ cm} \]

\[ \theta = \frac{3\pi}{4} \]

Substituting in the formula:

\[ 15\pi = r \times \frac{3\pi}{4} \]

\[ r = 15\pi \times \frac{4}{3\pi} \]

\[ r = 20 \text{ cm} \]

Hence, the correct option is:

(b) \(20\) cm