Here is the problem.
A cylinder is circumscribed about a sphere. If their volumes are denoted by $C$ and $S$, find $C$ as a function of $S$
My (Amended) Attempt:
[Based on the correction suggested by herbSteinberg]
Let $r$ be the radius of the sphere.
Then the height of the cylinder is $2r$, and the radius of the base is $r$. So the volume $C$ of the cylinder is given by $$ C = \pi r^2 (2r) = 2 \pi r^3. \tag{1} $$
And, the volume $S$ of the sphere is given by $$ S = \frac43 \pi r^3. \tag{2} $$
From (2), we obtain $$ r^3 = \frac{3}{4 \pi} S = \frac{ 3S }{4 \pi}, $$ and hence $$ r = \sqrt[3]{ \frac{3S}{4 \pi} }. \tag{3} $$
Finally, putting the value of $r$ from (3) into (1), we get $$ C = 2 \pi \left( \sqrt[3]{ \frac{3S}{4 \pi} } \right)^3 = 2 \pi \left( \frac{3S}{4 \pi} \right) = \frac32 S. $$
Is my solution correct in each and every detail? Or, are there any errors of approach or answer?
As noted in the comments, the radius of the cylinder is just $r$. Otherwise your work appears to be correct, but you've made things harder on yourself than necessary. Note that $r^3$ appears in both formulas, so once you have solved for $r^3 = \frac{3S}{4\pi}$, you can immediately use this expression in place of $r^3$ in the formula for $C$, going straight to the last equality.
In other words:
$$S = \frac{4}{3}\pi r^3 \Rightarrow r^3 = \frac{3S}{4\pi}$$ $$C = \pi r^2 \cdot (2r) = 2\pi r^3 = 2\pi\cdot\frac{3S}{4\pi}=\frac{3S}{2} $$