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 (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: