Substitute $r = 3$ into the hemisphere volume formula to get a specific number (with $\pi$) for its volume.

Write the cone’s volume using $r = 3$ and unknown height $h$, then set this expression equal to the hemisphere’s volume and solve for $h$.

When you have your equation, notice that both sides include $\pi$ and factors of $3$, which you can cancel to make solving for $h$ easier.

In Desmos, type V_hemi = (2/3)*pi*27 (since $3^3 = 27$) to get the hemisphere's volume. Type V_cone = (1/3)*pi*9*h as a function of height $h$.

Type y = (1/3)*pi*9*x and y = (2/3)*pi*27. Click on the intersection; the x-coordinate is the cone's height.

• Volume of a sphere with radius $r$ is $V = \frac{4}{3}\pi r^3$.
• A hemisphere is half a sphere, so its volume is $V = \frac{1}{2}\cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$.
• Volume of a cone with base radius $r$ and height $h$ is $V = \frac{1}{3}\pi r^2 h$.

We will use these with $r = 3$.

Substitute $r = 3$ into the hemisphere volume formula:

So the hemisphere has volume $18\pi$ cubic centimeters.

For the cone, the base radius is also $3$, so substitute $r = 3$ into the cone volume formula:

So the cone’s volume is $3\pi h$, where $h$ is the height we need to find.

The hemisphere and cone have the same volume, so set their volumes equal and solve for $h$:

Divide both sides by $3\pi$:

So the height of the cone is $6$ centimeters, which corresponds to choice D.