A circular hollow cone of height h and semi-vertical angle $\displaystyle 30^o$ stands on a horizontal table. An upright cylinder of radius $\displaystyle r$ cm and height $\displaystyle d$ cm just fits into the cone as shown in the diagram.

First part of the question: Show that $\displaystyle d=h-r\sqrt{3}$

Second part: Prove that the maximum possible volume of the cylinder is $\displaystyle \frac{4{\pi}h^3}{81}$

I completed the first part and am facing problems with the second part