To find the volume of the cylinder, we first need to find the radius of the sphere from its volume. 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 \( 48 , m^3 \), we can set up the equation:
\[ \frac{4}{3} \pi r^3 = 48 \]
To solve for \( r \), we first rearrange the equation:
\[ r^3 = \frac{48 \times 3}{4\pi} \]
Calculating this gives:
\[ r^3 = \frac{144}{4\pi} = \frac{36}{\pi} \]
Now we can take the cube root to find \( r \):
\[ r = \sqrt[3]{\frac{36}{\pi}} \]
Next, we will calculate the diameter \( d \) of the sphere:
\[ d = 2r = 2 \times \sqrt[3]{\frac{36}{\pi}} \]
Now that we have both the radius \( r \) for the cylinder and its height \( h \) (equal to the diameter of the sphere), we can write the volume formula for the cylinder. The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
We substitute \( h \) with \( 2r \):
\[ V = \pi r^2 (2r) = 2\pi r^3 \]
We know from earlier that \( \frac{4}{3} \pi r^3 = 48 \). Thus, we can find \( r^3 \):
\[ r^3 = \frac{48 \times 3}{4\pi} = \frac{144}{4\pi} = \frac{36}{\pi} \]
Now substituting \( r^3 \) into the volume formula for the cylinder:
\[ V = 2\pi \left(\frac{36}{\pi}\right) = 2 \times 36 = 72 , m^3 \]
Therefore, the volume of the cylinder is:
\[ \boxed{72} , m^3 \]