#### Question

Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .

#### Solution

Let:

Radius of the base = *r*,

Height = *h*,

Slant height = *l*,

Volume = *V*,

Curved surface area = *C*

\[\text { As, Volume }, V = \frac{1}{3}\pi r^2 h\]

\[ \Rightarrow h = \frac{3V}{\pi r^2}\]

\[\text { Also, the slant height, l } = \sqrt{h^2 + r^2}\]

\[ = \sqrt{\left( \frac{3V}{\pi r^2} \right)^2 + r^2}\]

\[ = \sqrt{\frac{9 V^2}{\pi^2 r^4} + r^2}\]

\[ = \sqrt{\frac{9 V^2 + \pi^2 r^6}{\pi^2 r^4}}\]

\[ \Rightarrow l = \frac{\sqrt{9 V^2 + \pi^2 r^6}}{\pi r^2}\]

\[\text { Now, }\]

\[\text { CSA, C } = \pi rl\]

\[ \Rightarrow C\left( r \right) = \pi r\frac{\sqrt{9 V^2 + \pi^2 r^6}}{\pi r^2}\]

\[ \Rightarrow C\left( r \right) = \frac{\sqrt{9 V^2 + \pi^2 r^6}}{r}\]

\[ \Rightarrow C'\left( r \right) = \frac{r \times \frac{6 \pi^2 r^5}{2\sqrt{9 V^2 + \pi^2 r^6}} - \sqrt{9 V^2 + \pi^2 r^6}}{r^2}\]

\[ = \frac{\left[ \frac{3 \pi^2 r^6 - \left( 9 V^2 + \pi^2 r^6 \right)}{\sqrt{9 V^2 + \pi^2 r^6}} \right]}{r^2}\]

\[ = \frac{3 \pi^2 r^6 - 9 V^2 - \pi^2 r^6}{r^2 \sqrt{9 V^2 + \pi^2 r^6}}\]

\[ = \frac{2 \pi^2 r^6 - 9 V^2}{r^2 \sqrt{9 V^2 + \pi^2 r^6}}\]

\[\text { For maxima or minima, C }'\left( r \right) = 0\]

\[ \Rightarrow \frac{2 \pi^2 r^6 - 9 V^2}{r^2 \sqrt{9 V^2 + \pi^2 r^6}} = 0\]

\[ \Rightarrow 2 \pi^2 r^6 - 9 V^2 = 0\]

\[ \Rightarrow 2 \pi^2 r^6 = 9 V^2 \]

\[ \Rightarrow V^2 = \frac{2 \pi^2 r^6}{9}\]

\[ \Rightarrow V = \sqrt{\frac{2 \pi^2 r^6}{9}}\]

\[ \Rightarrow V = \frac{\pi r^3 \sqrt{2}}{3} or \ r = \left( \frac{3V}{\pi\sqrt{2}} \right)^\frac{1}{3} \]

\[\text { So,} h = \frac{3}{\pi r^2} \times \frac{\pi r^3 \sqrt{2}}{3}\]

\[ \Rightarrow h = r\sqrt{2}\]

\[ \Rightarrow \frac{h}{r} = \sqrt{2}\]

\[ \Rightarrow \cot\theta = \sqrt{2}\]

\[ \therefore \theta = \cot^{- 1} \left( \sqrt{2} \right)\]

\[\text { Also }, \]

\[\text { Since, for } r < \left( \frac{3V}{\pi\sqrt{2}} \right)^\frac{1}{3} , C'\left( r \right) < 0 \text { and for } r > \left( \frac{3V}{\pi\sqrt{2}} \right)^\frac{1}{3} , C'\left( r \right) > 0\]

\[\text { So, the curved surface for r }= \left( \frac{3V}{\pi\sqrt{2}} \right)^\frac{1}{3} or V = \frac{\pi r^3 \sqrt{2}}{3}\text { is the least } .\]