The radius of the circle whose arc length $15\pi \,cm$ makes an angle of $\left( {\dfrac{{3\pi }}{4}} \right)$ radians at the centre is?

Hint: In this question we want to find the radius of a circle given the arc length and the angle subtended at the centre of the circle. We have a standard formula for finding the circumference of a circle when we are provided with the radius of the circle, $C = 2\pi r$. But we can find the length of arc using the formula: $\theta r$ , where $\theta $ is the measure of angle subtended by the arc at the centre of the circle in radians.

Complete step by step solution:
We have the arc length as $15\pi $ and the angle subtended by the same arc at the centre of the circle as $\left( {\dfrac{{3\pi }}{4}} \right)$ radians.
We have a formula for finding the length of an arc as $\theta r$.
Hence, we have, $15\pi \,cm = \left( {\dfrac{{3\pi }}{4}} \right)r$
Now, we have to find the value of radius from the above equation using the method of transposition.
So, shifting all the constant terms to the left side of the equation, we get,
$ \Rightarrow \dfrac{{15\pi }}{{\left( {\dfrac{{3\pi }}{4}} \right)}}\,cm = r$
Simplifying the expression, we get,
$ \Rightarrow r = 15\pi \times \left( {\dfrac{4}{{3\pi }}} \right)\,cm$
Cancelling the common terms in numerator and denominator, we get,
$ \Rightarrow r = 5 \times \left( {\dfrac{4}{1}} \right)\,cm$
Doing the calculations, we get,
$ \Rightarrow r = 20cm$
Therefore, the radius of the circle whose arc length $15\pi \,cm$ makes an angle of $\left( {\dfrac{{3\pi }}{4}} \right)$ radians at the centre is $20cm$.

Note: The formula for finding the arc length of an arc of a circle can be found using the unitary method. We know that the circumference of a circle that subtends an angle of $2\pi $ radians at the centre is $2\pi r$, where r is the radius of the circle. So, using the unitary method, the formula for finding the arc length of a circle is $\dfrac{\theta }{{2\pi }} \times 2\pi r = \theta r$.