To find the volume of the cone, we first need to determine the radius of the sphere using the volume formula for a sphere:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Given that the volume of the sphere is \( 72 , \text{m}^3 \):
\[ 72 = \frac{4}{3} \pi r^3 \]
To isolate \( r^3 \), we can multiply both sides by \( \frac{3}{4} \):
\[ r^3 = \frac{72 \times 3}{4 \pi} = \frac{216}{4 \pi} = \frac{54}{\pi} \]
Now, we can find the radius \( r \):
\[ r = \left( \frac{54}{\pi} \right)^{1/3} \]
Next, we need to calculate the height of the cone, which is equal to the diameter of the sphere:
\[ \text{Diameter} = 2r \]
Now we can use the formula for the volume of a cone:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Substituting \( h = 2r \):
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we can substitute \( r^3 = \frac{54}{\pi} \):
\[ V_{\text{cone}} = \frac{2}{3} \pi \left( \frac{54}{\pi} \right) = \frac{2 \times 54}{3} = \frac{108}{3} = 36 , \text{m}^3 \]
Thus, the volume of the cone is:
\[ \boxed{36 , \text{m}^3} \]