The height of the cylinder is given as exactly twice its radius. Rewrite the cylinder’s volume using $h = 2r$, and write the hemisphere’s volume using radius $r$.

Add the cylinder and hemisphere volumes to get the total volume in terms of $r$, then set this equal to $576\pi$. After that, divide out $\pi$ and solve the resulting equation for $r^3$.

Once you isolate $r^3$, think about what number cubed gives that value. You may find it helpful to test small integers like 4, 5, and 6.

In Desmos, type the equation (8/3)*pi*x^3 = 576*pi. Here x represents the radius $r$.

Either:

• Graph y1 = (8/3)*pi*x^3 and y2 = 576*pi and find the x-coordinate where the graphs intersect, or
• Use a table for y1 = (8/3)*pi*x^3 and look for the x-value where y1 equals 576*pi. The x-value that makes the equation true is the radius of the silo.

The silo is made of:

• A right circular cylinder of radius $r$ and height $2r$.
• A hemisphere of radius $r$ on top.

Volume formulas:

• Cylinder: $V_{\text{cyl}} = \pi r^2 h$.
• Sphere: $V_{\text{sphere}} = \frac{4}{3}\pi r^3$, so a hemisphere has volume $V_{\text{hemi}} = \frac{1}{2}\cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$. \nSubstitute $h = 2r$ into the cylinder volume: $V_{\text{cyl}} = \pi r^2 (2r) = 2\pi r^3$. \nSo the total volume in terms of $r$ is:

Add the two volume expressions:

The problem tells you the total volume is $576\pi$ cubic feet, so set this equal to $576\pi$:

First, divide both sides of the equation by $\pi$ (since $\pi \neq 0$):

Now isolate $r^3$ by multiplying both sides by $\frac{3}{8}$:

Compute $576 \cdot \frac{3}{8}$:

• $576 \div 8 = 72$.
• $72 \cdot 3 = 216$.

So $r^3 = 216$. The last step is to take the cube root.

We have $r^3 = 216$. Recognize that $216 = 6^3$ because $6 \cdot 6 \cdot 6 = 216$.

Therefore, $r = 6$.