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  4. The semi vertical angle of a right circular cone of maximum volume of a given slant height is

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Q. The semi vertical angle of a right circular cone of maximum volume of a given slant height is

Application of Derivatives

Solution:

$h=\ell \cos \theta$
$r =\ell \sin \theta$
$V =\frac{1}{3} \pi r ^2 h$
image
$V =\frac{1}{3} \pi \ell^3 \sin ^2 \theta \cos \theta$
$\frac{ dV }{ d \theta}=\frac{1}{3} \pi \ell^3\left(2 \sin \theta \cos ^2 \theta-\sin ^3 \theta\right)$
$\frac{ dV }{ d \theta}=\frac{1}{3} \pi \ell^3 \sin \theta\left(2-3 \sin ^2 \theta\right)=0$ at
$\sin \theta=\sqrt{\frac{2}{3}} \Rightarrow \tan \theta=\sqrt{2}$