To find the height of the cone, we can use the formula for the volume of a cone, which is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
- First, we need to find the radius of the cone. The diameter is given as 10 meters, so the radius \( r \) is:
\[ r = \frac{\text{diameter}}{2} = \frac{10}{2} = 5 \text{ meters} \]
- Now we have the volume \( V = 225\pi \) cubic meters and the radius \( r = 5 \) meters. We can substitute these values into the volume formula and solve for the height \( h \):
\[ 225\pi = \frac{1}{3} \pi (5^2) h \]
- Simplifying the right side gives:
\[ 225\pi = \frac{1}{3} \pi (25) h \]
\[ 225\pi = \frac{25}{3} \pi h \]
- Now, we can divide both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ 225 = \frac{25}{3} h \]
- To isolate \( h \), multiply both sides by 3:
\[ 675 = 25h \]
- Divide both sides by 25:
\[ h = \frac{675}{25} = 27 \]
Therefore, the height of the cone is:
\[ \boxed{27} \text{ meters} \]