To find the height of the cone, 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, and \( h \) is the height.
- Find the radius: The diameter of the cone is given as 10 meters, so the radius \( r \) is:
\[ r = \frac{\text{diameter}}{2} = \frac{10}{2} = 5 \text{ meters} \]
- Use the volume formula: We know the volume \( V \) is \( 225\pi \) cubic meters. Substitute \( V \) and \( r \) into the volume formula:
\[ 225\pi = \frac{1}{3} \pi (5^2) h \]
- Simplify: First, calculate \( 5^2 \):
\[ 5^2 = 25 \]
Then substitute it back into the volume equation:
\[ 225\pi = \frac{1}{3} \pi (25) h \]
- Cancel \( \pi \) from both sides:
\[ 225 = \frac{1}{3} (25) h \]
- Multiply both sides by 3 to eliminate the fraction:
\[ 675 = 25h \]
- Solve for \( h \):
\[ h = \frac{675}{25} = 27 \]
Thus, the height of the cone is:
\[ \boxed{27} \text{ meters} \]