The algebra is too much for me to put here, but I will outline the steps to take:
1. The length of the arc will be RQ, which will be the circumference of the cone base
2. The cone radius will be then r= QR/2*pi
3. The volume of a cone is hA/3, where h is the height and A is the base area.
4. Calculate A from the cone radius= pi*r^2= Q*Q*R*R/4*pi
5. Calc the height of the cone from h^2= r*r + R*R
6. Now calc the cone volume as a function of R:
V(R)= (1/3)hA= (1/3) (√r*r + R*R) Q*Q*R*R/(4*pi)
now put in for and you will have V+V(R), which you can differentiate wrt R, and find the extrema, if there is one. I think there will be an extrema somewhere between Q= 0 and Q= 2*pi, maybe at pi? Not sure. It's a good calculus problem, but maybe a little bit of algebra.