Question

Show that semi-vertical angle of right circular cone of given surface area and maximum volume is $$\sin ^{ -1 }{ \left( \cfrac { 1 }{ 3 } \right) }$$.

Solution

$$h=r\text{cot}\theta$$surface area is constant;$$S=\pi r^{3}(1+\text{cosec}\theta)$$$$\displaystyle \frac{dS}{dr}=0\implies\frac{d\theta}{dr}=\frac{2(1+\text{cosec}\theta)}{r(\text{cosec}\theta\text{cot}\theta)}$$volume to be maximum ;$$\displaystyle V=\frac{\pi r^{3}\text{cot}\theta}{3}\implies \frac{dV}{dr}=\frac{r^{2}\bigg(3\text{cot}\theta-r\text{cosec}^{2}\theta\displaystyle \frac{d\theta}{dr}\bigg)}{3}=0 \implies x=\text{sin}^{-1}\bigg(\frac{1}{3}\bigg)$$Maths

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