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 is one-third that of the cone and the greatest volume of cylinder is `4/(27)pih^3tan^2alphadot`

Explanation

To solve the problem of finding the height of the cylinder of greatest volume that can be inscribed in a right circular cone of height \( h \) and semi-vertical angle \( \alpha \), we will follow these steps:

### Step 1: Understand the Geometry
We have a right circular cone with height \( h \) and a semi-vertical angle \( \alpha \). The radius of the base of the cone can be expressed in terms of \( h \) and \( \alpha \) as:
\[ R = h \tan \alpha \]
where \( R \) is the radius of the cone's base.
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