Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is `cot^(-1)(sqrt(2))dot`

Explanation

Let radius, height and semi-vertical angle of cone be `r,h` and `theta` respectively.

Volume,`V=1/3pir^2h`

`implies h={3V}/{pir^2}`

Curved surface area is,

`A=pirsqrt{r^2+h^2}`

`implies A=sqrt{pi^2r^4+{9V^2}/{r^2}}`

`{dA}/{dr}=1/ sqrt{pi^2r^4+{9V^2}/{r^2}}(4pi^2r^3-{18V^2}/r^3)`

Put `{dA}/{dr}=0`

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