1. Set up the geometry

Let the sphere have radius $R=3$.

Suppose a right circular cylinder is inscribed in the sphere with:

• radius $r$
• height $h$

If we take the center of the sphere as the origin, then the top edge of the cylinder lies on the sphere. From the right triangle formed,

$r^2 + \left(\frac h2\right)^2 = R^2 = 9.$

So,

$r^2 = 9 - \frac{h^2}{4}.$

2. Write the volume of the cylinder

Volume of the cylinder is

$V = \pi r^2 h.$

Substitute $r^2$:

$V(h) = \pi \left(9 - \frac{h^2}{4}\right)h.$

$V(h) = 9\pi h - \frac{\pi}{4}h^3.$

3. Maximize the volume

Differentiate with respect to $h$:

$\frac{dV}{dh} = 9\pi - \frac{3\pi}{4}h^2.$

For maximum volume, set $\frac{dV}{dh}=0$:

$9\pi - \frac{3\pi}{4}h^2 = 0.$

Divide by $3\pi$:

$3 - \frac{h^2}{4} = 0.$

$\frac{h^2}{4} = 3.$

$h^2 = 12.$

$h = 2\sqrt{3}.$

Since height is positive, we take the positive value.

4. Check that it is a maximum

$\frac{d^2V}{dh^2} = -\frac{3\pi}{2}h.$

At $h=2\sqrt{3}$,

$\frac{d^2V}{dh^2} < 0,$

so the volume is indeed maximum.

5. Match with options

The height is

$\boxed{2\sqrt{3}}.$

So the correct option is B.