Question

A right circular cylinder is inscribed in a sphere of radius R.
Show that the volume is maximum when its height is 2R/√3.​

Answer 1

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is \(\large\frac{2R}{\sqrt 3}\) .

A)

Toolbox:

Volume =πr2h

Step 1:

Radius of the sphere=R

Let h be the diameter of the base of the inscribed cylinder .

Then,

h2+x2=(2R)2

h2+x2=4R2------(1)

Volume of the cylinder =πr2h

V=π(x22)2.h

=πx44.h

Volume=14πx2h

Substituting the value of x2 we get

V=14πh(4r2−h2)

From (1),x2=4R2−h2

V=πR2h−14πh3