Question

Find the maximum volume of the cylinder which can be inscribed in a sphere of radius $3\sqrt{3}$cm. (Leave the answer in terms of $\pi$).

Answer 1

Let $h$ units be the height of the cylinder and $R$ unit be the radius of the cylinder. Given $3\sqrt{3}$ cm is the radius of the sphere.
If $V$ is the volume of the cylinder, then
$V=\pi R^{2}h$ ..... (i)

Let O be the centre of the sphere and C'OC $\perp$ B'A' as well as BA.
From right angled $\Delta OCA$,
$(3\sqrt{3}{)}^2={\left(\frac{h}{2}\right)}^2+R^2$
$\Rightarrow R^2=27-\frac{h^2}{4}$ ...... (ii)
$\therefore V=\pi (27-\frac{h^2}{4}).h$
$V=27\pi h-\pi \frac{h^3}{4}$
$\therefore \frac{dV}{dh}=27\pi -\frac{3\pi h^2}{4}$
and

$\frac{d^{2}V}{d{h}^2}=-\frac{6\pi h}{4}$

For Maxima/Minima, $\frac{dV}{dh}=0$
$\therefore 27\pi =\frac{3\pi h^2}{4}$
$h^2=\frac{4\times 27}{3}$
$h^2=36$
$h=6,-6$
and

${\left(\frac{d^{2}V}{d{h}^2}\right)}_{h=6}=\frac{-6\pi \times 6}{4}<0$

$\therefore$ V is maximum when h = 6, putting h = 6 in equation (ii)
$R^2=27-\frac{36}{4}$
$R^2=18$
$\therefore$ From (i) $V=\pi R^{2}h$
$=\pi \times 18\times 6$
$=108\pi c{m}^3$