Question

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

• A. ${\cos }^{-1}\sqrt{2}$
• B. ${\sin }^{-1}\sqrt{2}$
• C. ${\tan }^{-1}\sqrt{3}$
• D. ${\tan }^{-1}\sqrt{2}$

Answer 1

Answer: (D)

$h=\ell \cos \theta$
$r=\ell \sin \theta$
$V=\frac{1}{3}\pi r^{2}h$

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