To find the volume of the cylinder, we first need to determine the radius of the sphere using its volume.
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere. Given that the volume of the sphere is \( 48 , m^3 \), we can set up the equation:
\[ \frac{4}{3} \pi r^3 = 48 \]
To solve for \( r^3 \), we can rearrange the equation as follows:
\[ r^3 = \frac{48 \cdot 3}{4 \pi} = \frac{144}{4 \pi} = \frac{36}{\pi} \]
Next, we take the cube root to find \( r \):
\[ r = \sqrt[3]{\frac{36}{\pi}} \]
Now we can calculate the diameter of the sphere, which is:
\[ d = 2r = 2\sqrt[3]{\frac{36}{\pi}} \]
Next, we need to find the volume of the cylinder. The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( h \) is the height of the cylinder. Since the height of the cylinder is equal to the diameter of the sphere, we use the diameter we found:
\[ h = d = 2r \]
Now we substitute this into the volume formula of the cylinder:
\[ V = \pi r^2 (2r) = 2\pi r^3 \]
Substituting \( r^3 \) from earlier:
\[ V = 2\pi \left(\frac{36}{\pi}\right) = 2 \cdot 36 = 72 , m^3 \]
Thus, the volume of the cylinder is:
\[ \boxed{72 , m^3} \]