First slide

Maxima and minima

Question

The height of the right circular cylinder of maximum volume that can be inscribed in a sphere of radius ‘a’ is

Moderate

A

$\frac{a}{\sqrt{3}}$
B

$\frac{2 a}{\sqrt{3}}$
C

$\frac{\sqrt{2}}{a}$
D

$a \sqrt{2}$

Solution

Volume of the right circular cylinder
$V = π r^{2} h = π h (a^{2} - \frac{h^{2}}{4}) (∵ a^{2} = r^{2} + \frac{h^{2}}{4})$
$= π (a^{2} h - \frac{h^{3}}{4})$
$\begin{aligned} Differentiate with respect to ' h' \\ \Rightarrow \frac{d v}{d h} = π (a^{2} - \frac{3 h^{2}}{4}) = 0 \Rightarrow h = \frac{2 a}{\sqrt{3}} \end{aligned}$

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