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 is

The given right circular cone of fixed height (h) and semi vertical angle ().

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 get,

r = h tanα

Now, ΔAOG is similar to ΔCEG, we have:

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

Now, if , then,

Now,

Then, by second derivative test, the volume of the cylinder is the greatest when

So, When, H =

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

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

Hence Proved.