question_answer

An arc of a circle is of length 5n cm and the sector it bounds has an area of \[20\,\pi \,c{{m}^{2}}\], then radius of the circle is

A) 4cm

B) 8cm

C) 12cm

D) 16cm

Correct Answer: B

Solution :

Let the radius of the circle be r cm and the arc AB of length \[5\pi \]cm subtends angle \[\theta \] at the centre O of the circle.   Then, arc \[AB=5\pi \] cm and area of the sector \[OAB=20\pi \,c{{m}^{2}}\] \[\Rightarrow \,\,\frac{\theta }{360{}^\circ }\times 2\pi r=5\pi \] And \[\frac{\theta }{360{}^\circ }\times \pi {{r}^{2}}=20\pi \] \[\Rightarrow \,\,\frac{\frac{\theta }{360{}^\circ }\times \pi {{r}^{2}}}{\frac{\theta }{360{}^\circ }\times 2\pi r}=\frac{20\pi }{5\pi }\] \[\Rightarrow \frac{r}{2}=4\,\,\Rightarrow r=8\,cm\]