The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Given that the volume of the sphere is \( 72 , \text{m}^3 \), we can set up the equation:
\[ 72 = \frac{4}{3} \pi r^3 \]
To find the radius \( r \), we can solve for \( r^3 \):
\[ r^3 = \frac{72 \times 3}{4\pi} = \frac{216}{4\pi} = \frac{54}{\pi} \]
Now, we calculate the radius:
\[ r = \left( \frac{54}{\pi} \right)^{1/3} \]
Next, we need to calculate the volume of the cone, which is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
The height \( h \) of the cone is equal to the diameter of the sphere, which is \( 2r \):
\[ h = 2r \]
Now substituting \( h \) into the volume formula for the cone:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
We know \( r^3 = \frac{54}{\pi} \), so substituting that into the volume of the cone gives:
\[ V_{\text{cone}} = \frac{2}{3} \pi \left( \frac{54}{\pi} \right) = \frac{2 \times 54}{3} = \frac{108}{3} = 36 \]
Thus, the volume of the cone is \( 36 , \text{m}^3 \).
So, the final answer is:
\[ \text{Volume of the cone} = 36 , \text{m}^3 \]
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