A spherical water tank and a right circular cylindrical container both have the exact same radius of $r$. If the height of the cylindrical container is equal to $3r$, what is the ratio of the volume of the cylinder to the volume of the sphere?

The correct option is 9:4.

To find the ratio of the volume of the cylindrical container to the volume of the spherical water tank, we calculate their respective volumes in terms of radius $r$.

Step 1: Formula for the volume of the sphere
The volume of a sphere of radius $r$ is:
$V_{\text{sphere}}=\frac{4}{3}\pi r^3$

Step 2: Formula for the volume of the cylinder
The volume of a right circular cylinder with radius $r$ and height $h$ is:
$V_{\text{cylinder}}=\pi r^{2}h$
Given that the height of the cylindrical container is $h=3r$, we substitute this into the formula:
$V_{\text{cylinder}}=\pi r^2\left(3r\right)=3\pi r^3$

Step 3: Calculating the ratio
Now, we calculate the ratio of the volume of the cylinder to the volume of the sphere:
$\text{Ratio}=\frac{V_{\text{cylinder}}}{V_{\text{sphere}}}=\frac{3\pi r^3}{\frac{4}{3}\pi r^3}$
Canceling out the common factor $\pi r^3$ from numerator and denominator:
$\text{Ratio}=\frac{3}{\frac{4}{3}}=3\times \frac{3}{4}=\frac{9}{4}$

Therefore, the ratio of the volume of the cylinder to the volume of the sphere is 9:4.