Hello I am trying to solve for the smallest possible radius $r$ for a sphere that can inscribe a circular cylinder of volume 8 cubic units(so the volume is given and is a constant 8 cubic units). I have already solved for a formula which gives the maximum possible volume of a circular cylinder dependent of the radius $r$ of the sphere with
$V_{max of cylinder} = \frac{4\pi r^3}{3\sqrt{3}}$
Returning to the new problem of finding the smallest radius $r$ that can inscribe a circular cylinder of 8 cubic units(but not necessarily a circular cylinder that has an optimal volume which follows the above formula), how can one apply the Arithmetic Geometric Mean Inequality to minimize the radius $r$ of the sphere which inscribes any circular cylinder of volume 8 cubic units? I apologize if the way I stated this problem causes ambiguity, so I will just assume the problem I am really trying to solve is "What is the smallest possible radius $r$ for a sphere that can inscribe a circular cylinder of 8 cubic units?" and that really under these conditions, one may assume that the cylinder itself has an optimal maximum volume following the formula I stated above. For educational purposes, I want to know how to exactly set up problem of applying the Arithmetic Geometric Mean Inequality for minimizing the radius of the sphere which can enclose the circular cylinder of 8 cubic units.
In addition, how would a single-variable calculus approach to finding the minimum radius follow? The formula above I obtained for the maximum possible volume of an inscribed circular cylinder in a sphere for a given sphere radius $r$ was obtained through single variable calculus(the forumla gives the maximum possible volume of a cylinder for a given sphere radius), I want to know if there is a way to similarly apply a single variable calculus approach to finding the minimum radius $r$ that can enclose a known volume for an inscribed circular cylinder. Any help on either questions I have stated would be appreciated, thanks.