To find the height of the cylinder, we can use the formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height.
We know the following values:
- The volume \( V = 441\pi \) mm³,
- The radius \( r = 7 \) mm.
We can substitute these values into the formula:
\[ 441\pi = \pi (7^2) h \]
First, calculate \( 7^2 \):
\[ 7^2 = 49 \]
Now substitute this back into the equation:
\[ 441\pi = \pi (49) h \]
To isolate \( h \), divide both sides by \( \pi \):
\[ 441 = 49h \]
Next, solve for \( h \) by dividing both sides by 49:
\[ h = \frac{441}{49} \]
Now calculate \( \frac{441}{49} \):
\[ h = 9 \]
Thus, the height of the cylinder is:
\[ \boxed{9 \text{ mm}} \]