## Maximum Volume of Cylinder Inscribed In Sphere

What is the maximum volume of a cylinder inscribed in a sphere of radius
$R$
?

$r$
of the cylinder base subtend an angle of
$\theta$
at the centre of the sphere then
$r=Rsin \theta$
and if the height of the cylinder is
$h$
then from the diagram
$h/2=R cos \theta \rightarrow h=2Rcos \theta$
.
The volume of the cylinder is
$V= \pi r^2h=\pi (R sin \theta )^2 (2R cos \theta )=2R^3 sin^2 \theta cos \theta$

$\frac{dV}{d \theta }= 2 \pi R^3 (2 sin \theta cos^2 \theta -sin^3 \theta )$

Then for maximum volume
$2 sin \theta cos \theta -sin^3 \theta = sin \theta (2cos^2 \theta -sin^2 \theta )=0 \rightarrow tan^2 \theta =2$

$sin^2 \theta =1- cos^2 \theta = 1- \frac{1}{sec^2 \theta }=1- \frac{1}{1+tan^2 \theta }=1- \frac{1}{1+2}=\frac{2}{3}$
.
$cos \theta = \sqrt{1-sin^2 \theta }= \sqrt{1-\frac{2}{3}}= \frac{\sqrt{3}}{3}$

Then
$V=2\pi R^3 \frac{2}{3} \frac{1}{\sqrt{3}}= \frac{4 \pi R^3 \sqrt{3}}{9}$
.