A cylinder is circumscribed about a sphere. If their volumes are denoted by $C$ and $S$, find $C$ as a function of $S$

Question

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?

Answer 1

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:

$$S = \frac{4}{3}\pi r^3 \Rightarrow r^3 = \frac{3S}{4\pi}$$ $$C = \pi r^2 \cdot (2r) = 2\pi r^3 = 2\pi\cdot\frac{3S}{4\pi}=\frac{3S}{2} $$

Answer 2

The radius of the sphere should be equal to the radius of the cylinder face. Though you are correct about the height of the cylinder being twice the radius of the sphere.