Show that height of the cylinder of
greatest volume which can be inscribed in a right circular cone of
height *h* and semi vertical angle *α*
is one-third that of the cone and the greatest volume of cylinder
istan^{2} *α*.

The given
right circular cone of fixed height (*h)* and semi-vertical
angle (*α**)* can
be drawn as:

Here, a
cylinder of radius *R* and height *H* is inscribed in the
cone.

Then, ∠GAO
= *α*, OG =* r*,
OA = *h*, OE = *R*, and CE = *H*.

We have,

*r *=
*h* tan *α*

Now, since ΔAOG is similar to ΔCEG, we have:

Now, the
volume (*V)* of the cylinder is given by,

And, for, we have:

∴By second derivative test, the volume of the cylinder is the greatest when

Thus, the height of the cylinder is one-third the height of the cone when the volume of the cylinder is the greatest.

Now, the maximum volume of the cylinder can be obtained as:

Hence, the given result is proved.

**
**