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 of the base,
- \( h \) is the height.
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First, we need to find the radius from the diameter: \[ \text{Diameter} = 10 \text{ meters} \implies r = \frac{\text{Diameter}}{2} = \frac{10}{2} = 5 \text{ meters} \]
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Now we substitute the volume and radius into the volume formula. We know that the volume \( V = 225\pi \): \[ 225\pi = \frac{1}{3} \pi (5^2) h \] Simplifying this, we get: \[ 225\pi = \frac{1}{3} \pi (25) h \] \[ 225\pi = \frac{25}{3} \pi h \]
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We can cancel \(\pi\) from both sides: \[ 225 = \frac{25}{3} h \]
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Next, we multiply both sides by 3 to eliminate the fraction: \[ 675 = 25h \]
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Finally, divide both sides by 25 to solve for \( h \): \[ h = \frac{675}{25} = 27 \text{ meters} \]
Thus, the height of the cone is \( \boxed{27} \) meters.