Question

Find the volume of the largest cylinder that can be inscribed in a sphere of radius r.

OR,

Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is \(\frac{2R}{\sqrt3}.\). Also find the maximum volume.

Answer 1

Let R, h be the radius and height of inscribed cylinder respectively.

If V be the volume of cylinder then

V = πR²h

Differentiating with respect to h, we get

\(\frac{dV}{dh}=\) \(\pi(r^2-\frac{3h^2}{4})\) ....(i)

For maxima or minima

\(\frac{dV}{dh}=\) 0

⇒ \(\pi(r^2-\frac{3h^2}{4})\) = 0

⇒ \(r^2-\frac{3h^2}{4}=0\)

⇒  \(r=\frac{h\sqrt{3}}{2}\)

⇒ \(h=\frac{2r}{\sqrt{3}}\)

Differentiating (i) again with respect to h, we get