Substitute $r=9$ into the hemisphere volume formula and simplify to get a numerical expression (with $\pi$) for its volume. Then write the cylinder’s volume in terms of its unknown height $h$ using $r=6$.

Because the two solids have the same volume, set the hemisphere volume equal to the cylinder volume and solve the resulting equation for $h$.

When you solve for $h$, cancel common factors (like $\pi$) from both sides before dividing. Carefully simplify the fraction you get for $h$.

In Desmos, type (2/3)*pi*9^3 and note the numerical result. This is the hemisphere’s volume in cubic centimeters.

In a new line, type ((2/3)*pi*9^3)/(pi*6^2). This expression represents the hemisphere’s volume divided by the cylinder’s base area $\pi(6)^2$, which gives the cylinder’s height.

Look at the value Desmos outputs for ((2/3)*pi*9^3)/(pi*6^2). That number is the required height of the cylinder in centimeters.

For a sphere of radius $r$, the volume is $V = \frac{4}{3}\pi r^3$. A hemisphere is half of a sphere, so its volume is

For a right circular cylinder with radius $r$ and height $h$, the volume is

The hemisphere has radius $9$, so plug $r=9$ into the hemisphere formula:

Compute $9^3 = 729$, so

Now compute $\frac{2}{3}\cdot 729 = 486$, so the hemisphere’s volume is $486\pi$ cubic centimeters.

The cylinder has radius $6$, so its volume is

We are told the cylinder has the same volume as the hemisphere, so set the volumes equal:

Solve the equation

by first canceling $\pi$ from both sides:

Now divide both sides by $36$:

So the height of the cylinder is $\dfrac{27}{2}$ centimeters, which corresponds to choice C.