Substitute $r = 6$ into the hemisphere volume formula and simplify to get a numerical value (times $\pi$) for the hemisphere’s volume.

Add the expression for the cylinder’s volume and the numerical value for the hemisphere’s volume, set this equal to $432\pi$, then solve the resulting linear equation for the cylinder’s height $h$.

In Desmos, define the total volume expression for the solid as a function of the cylinder’s height, for example: f(x) = 36*pi*x + 144*pi. Here, x represents the height of the cylindrical portion.

On a new line, enter the constant function g(x) = 432*pi. This represents the given total volume of the solid.

Look at the graph and find the intersection point of the lines y = f(x) and y = g(x). The x-coordinate of this intersection is the cylinder’s height in units; read that value from the graph or from the intersection point details.

The solid is made of:

• A right circular cylinder with radius $r=6$ and unknown height $h$.
• A hemisphere (half of a sphere) with the same radius $r=6$.

Volume formulas:

• Cylinder: $V_{\text{cyl}} = \pi r^2 h$.
• Sphere: $V_{\text{sphere}} = \tfrac{4}{3}\pi r^3$, so a hemisphere has volume $V_{\text{hemi}} = \tfrac{1}{2} \cdot \tfrac{4}{3}\pi r^3 = \tfrac{2}{3}\pi r^3$.

Use $r = 6$ in the hemisphere volume formula:

Compute $\tfrac{2}{3} \cdot 216 = 144$, so

The total volume is the sum of the cylinder and hemisphere volumes, and we are told this equals $432\pi$.

First write the cylinder’s volume with $r = 6$:

Now set total volume equal to $432\pi$:

Divide every term by $\pi$ (since $\pi \neq 0$) to simplify:

Solve the linear equation $36h + 144 = 432$:

• Subtract $144$ from both sides:

• Divide both sides by $36$:

So, the height of the cylindrical portion is $8$ units, which corresponds to answer choice B.