To find the volume of the cone, we first need to determine the radius and diameter of the sphere from its volume.
The formula for the volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( V \) is the volume and \( r \) is the radius.
Given that the volume of the sphere is 72 m³, we can set up the equation:
\[ \frac{4}{3} \pi r^3 = 72 \]
To solve for \( r \):
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Multiply both sides by \( \frac{3}{4} \): \[ \pi r^3 = 72 \cdot \frac{3}{4} = 54 \]
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Divide by \( \pi \): \[ r^3 = \frac{54}{\pi} \]
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Take the cube root of both sides to find \( r \): \[ r = \sqrt[3]{\frac{54}{\pi}} \]
Next, we need to calculate the diameter of the sphere, which is:
\[ d = 2r \]
The formula for the volume of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( h \) is the height of the cone, and since the height is equal to the diameter of the sphere, we can substitute \( h \) with \( 2r \):
Thus, the volume of the cone becomes:
\[ V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now substituting \( r^3 \) from the earlier calculation:
\[ V = \frac{2}{3} \pi \left(\frac{54}{\pi}\right) \]
Simplifying further:
\[ V = \frac{2 \times 54}{3} = \frac{108}{3} = 36 \]
Therefore, the volume of the cone is \( \boxed{36 , m^3} \).
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