2015 NYJC P1 Q6

The diagram shows a right circular cone inscribed in a sphere of fixed radius \(R\) cm. The cone has a base radius of \(r\) cm and a height of \(h\) cm.

1. Show that \({{r}^{2}}=2Rh-{{h}^{2}}\).[1]
2. Use differentiation to find, in terms of \(R\), the value of \(h\) which gives a maximum volume of the cone.[5]
3. Find also the exact ratio of the base radius to the height of the cone, \(\frac{r}{h}\), in this case, simplifying your answer.[2]