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

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Solution

$h = r cot θ$
surface area is constant;
$S = π r^{3} (1 + cosec θ)$
$\frac{d S}{d r} = 0 ⟹ \frac{d θ}{d r} = \frac{2 (1 + cosec θ)}{r (cosec θ cot θ)}$
volume to be maximum ;
$V = \frac{π r^{3} cot θ}{3} ⟹ \frac{d V}{d r} = \frac{r^{2} (3 cot θ - r {cosec}^{2} θ \frac{d θ}{d r})}{3} = 0 ⟹ x = {sin}^{- 1} (\frac{1}{3})$

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ℓ

{tan}^{- 1} \sqrt{2}

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