In the figure, the cone has radius $r$, height $h$ and slant angle $\theta$, and the inscribed sphere has radius $R$ and center $O$.
The problem has 2 questions:
$a.$ $\space$ Compute the volume of the cone, in function of $\theta$
$b.$ $\space$ Which slant angle should be used to minimize the volume of the cone?
My try
For $a:$
It's easy to see that $HODC$ is a deltoid, so $$tan(\frac{θ}{2})=\frac{R}{r}$$ or $$r=\frac{R}{tan(\frac{θ}{2})}$$
and $$tan(θ)=\frac{h}{r}$$ or $$r\cdot tan(θ)=h$$ or $$\frac{R}{tan(\frac{θ}{2})}\cdot tan(θ)=h$$
Using this $r$ and $h$ in the formula for the volume of the cone:
$$V_{cone}=\frac{\pi R^3 tan(θ)}{3\cdot tan^3(\frac{θ}{2})}$$
but after this i don't see how to proceed to express $R$ in function of θ.
For $b$, i think i just need to find the final expression for $a$, and minimize that expression, because in $b$, $R$ is constant.
Any hints?