Question

The ratio of total surface area and lateral surface area of cone is 3:2 and its height is 6 meters. Find its volume of the cone ?

Answer 1

(A) Total surface Area of a cone =$\pi rl+\pi r^2$
$=\pi r\sqrt{r^2+h^2}+\pi r^2$
(LSA) Loteral surface area of cone =$\pi rl$
$=\pi r\sqrt{r^2+h^2}$
Given $\Rightarrow \frac{TSA}{LSA}=\frac{3}{2}\Rightarrow \frac{\pi r\sqrt{r^2+h^2}+\pi r^2}{\pi r\sqrt{r^2+h^2}}=\frac{3}{2}$
$\frac{3}{2}=1+\frac{r}{\sqrt{r^2+6^2}}$
$\frac{3}{2}-1=\frac{r}{\sqrt{r^2+36}}$
$\frac{1}{2}=\frac{r}{\sqrt{r^2+36}}$
$\frac{1}{4}=\frac{r^2}{r^2+36}$
$r^2+36=4r^2$
$36=3r^2$
$12=r^2$
$r=\sqrt{12}=2\sqrt{3}$
Volume $=\frac{1}{3}\pi r^{2}h$
$=\frac{1}{3}\pi \times (2\sqrt{3}{)}^2\times 6$
=$\frac{1}{3}\pi \times 4\times 3\times 6$
=24$\pi$