Sorry mistakes were made before! I realised after reading the following posting from tibol.
Working out r=R[3*sqrt(3)-1] gives r=4.196152423*R so the radius of the cylinder would be four times greater than that of the smaller sphere. This seems too large.
The triangle that is inscribed in the larger circle is equilateral - a fact that can be used to make progress. Each angle is 60 degrees
Try drawing the radius so that it bisects one of these angles.
The hypotenuse of the triangle formed is of length y, the adjacent side is of length R. Angle is half of 60 or 30.
| \ y
C R B
Now cos(30)=(sqrt(3)) / 2) = Adj / Hyp = y / R
sqrt(3)/2 = R/ y
y = 2/(sqrt(3) * R
y = 1.1547 * R
r=1.1547R + R
Adding on the the radius of the smaller circle R gives r=2.1
There are probably a few ways of doing this problem. I'll be the first to admit that I've assumed a lot about the radius of the cylinder bisects the angles of the triangle neatly!