To find the height of a cone given its volume and diameter, we can use the formula for the volume of a cone:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height.
Given:
- The volume \( V = 225\pi \) cubic meters,
- The diameter is 10 meters, so the radius \( r \) is half of the diameter:
\[ r = \frac{10}{2} = 5 \text{ meters} \]
Now, we can substitute \( V \) and \( r \) into the volume formula and solve for \( h \):
\[ 225\pi = \frac{1}{3} \pi (5^2) h \]
This simplifies to:
\[ 225\pi = \frac{1}{3} \pi (25) h \]
By cancelling \( \pi \) from both sides, we get:
\[ 225 = \frac{25}{3} h \]
Now, multiply both sides by 3 to eliminate the fraction:
\[ 675 = 25h \]
Now, divide both sides by 25 to solve for \( h \):
\[ h = \frac{675}{25} = 27 \]
Thus, the height of the cone is:
\[ \boxed{27} \text{ meters} \]