If the area of a sector of a circle bounded by an arc
Question:

If the area of a sector of a circle bounded by an arc of length 5π cm is equal to 20π cm2, then its radius is

(a) 12 cm

(b)16 cm

(c) 8 cm

(d) 10 cm

Solution:

We have given length of the arc and area of the sector bounded by that arc and we are asked to find the radius of the circle.

We know that area of the sector $=\frac{\theta}{360} \times \pi r^{2}$.

Length of the arc $=\frac{\theta}{360} \times 2 \pi r$

Now we will substitute the values.

Area of the sector $=\frac{\theta}{360} \times \pi r^{2}$

$20 \pi=\frac{\theta}{360} \times \pi r^{2}$……..(1)

Length of the $\operatorname{arc}=\frac{\theta}{360} \times 2 \pi r$

$5 \pi=\frac{\theta}{360} \times 2 \pi r$……….(2)

Now we will divide equation (1) by equation (2),

$\frac{20 \pi}{5 \pi}=\frac{\frac{\theta}{360} \times \pi r^{2}}{\frac{\theta}{360} \times 2 \pi r}$

Now we will cancel the like terms.

$\frac{20}{5}=\frac{r^{2}}{2 r}$

$\therefore 4=\frac{r}{2}$

$\therefore r=8$

Therefore, radius of the circle is $8 \mathrm{~cm}$.

Hence, the correct answer is option $(c)$.