To find the volume of the cone, we first need to determine the radius of the sphere using its volume formula. The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Here, we know that the volume \( V = 72 , m^3 \). We can set up the equation:
\[ 72 = \frac{4}{3} \pi r^3 \]
Now, we solve for \( r^3 \):
\[ r^3 = \frac{72 \cdot 3}{4\pi} = \frac{216}{4\pi} = \frac{54}{\pi} \]
Next, we find the radius \( r \):
\[ r = \sqrt[3]{\frac{54}{\pi}} \]
Now that we have \( r \), we need to determine the height of the cone. The diameter of the sphere is twice the radius:
\[ \text{Height of the cone} = \text{Diameter of the sphere} = 2r \]
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting \( h = 2r \) into the cone's volume formula gives us:
\[ V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we can substitute \( r^3 \) from our previous calculation:
\[ V = \frac{2}{3} \pi \left(\frac{54}{\pi}\right) \]
Simplifying this:
\[ V = \frac{2 \cdot 54}{3} = \frac{108}{3} = 36 , m^3 \]
Thus, the volume of the cone is:
\[ \boxed{36 , m^3} \]
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