Question

The area of the base of a right circular cone is 154 cm² and its height is 14 cm. Its curved surface area is
(a) $154\sqrt{5}{\mathrm{cm}}^2$
(b) $154\sqrt{7}{\mathrm{cm}}^2$
(c) $77\sqrt{7}{\mathrm{cm}}^2$
(d) $77\sqrt{5}{\mathrm{cm}}^2$

Answer 1

(a) $154\sqrt{5}{\mathrm{cm}}^2$
Area of the base of the of a right circular cone$=\mathrm{\pi }{r}^2$
Therefore,
$$\begin{gathered} \mathrm{\pi }{r}^2=154 \\ \Rightarrow \frac{22}{7}\times r^2=154 \\ \Rightarrow r^2=\left(154\times \frac{7}{22}\right) \\ \Rightarrow r^2=49 \\ \Rightarrow r=7\text{cm} \end{gathered}$$
Now, r = 7 cm and h = 14 cm
Then, slant height of the cone, $l=\sqrt{r^2+h^2}$
$$\begin{gathered} =\sqrt{{\left(7\right)}^2+{\left(14\right)}^2} \\ =\sqrt{49+196} \\ =\sqrt{245} \\ =7\sqrt{5}\text{cm} \end{gathered}$$
Hence, the curved surface area of the cone$=\mathrm{\pi }rl$
$$\begin{gathered} =\left(\frac{22}{7}\times 7\times 7\sqrt{5}\right){\text{cm}}^2 \\ =154\sqrt{5}{\text{cm}}^2 \end{gathered}$$