Originally Posted by

**galactus** These mostly ask for the greatest volume. This is a change.

Let r=radius of cone, R=radius of sphere, h=height of cone.

$\displaystyle \text{surface area of cone}={\pi}r\sqrt{r^{2}+h^{2}}$

By using triangles, we can see that $\displaystyle h=y+R$

Where, by triangle OAB, $\displaystyle y=\sqrt{R^{2}-r^{2}}$

So, $\displaystyle h=\sqrt{R^{2}-r^{2}}+R$

Now, R is a constant, so we have it down to one variable, r, the radius of the cone.

$\displaystyle S={\pi}r\sqrt{r^{2}+(\sqrt{R^{2}-r^{2}}+R)^{2}}$

This produces a booger to differentiate, though.

Doing this, I get $\displaystyle r\approx{0.85R}, \;\ or \;\ .525R$

Have to check to see which gives the max.