To find the volume of the cylinder, we first need to determine the radius and height of the cylinder based on the volume of the sphere.
The formula for the volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
We know the volume \( V \) of the sphere is \( 48 , \text{m}^3 \). We can set up the equation:
\[ \frac{4}{3} \pi r^3 = 48 \]
Now, solve for \( r^3 \):
\[ r^3 = \frac{48 \times 3}{4 \pi} \]
\[ r^3 = \frac{144}{4 \pi} = \frac{36}{\pi} \]
Now, let's find \( r \):
\[ r = \left(\frac{36}{\pi}\right)^{\frac{1}{3}} \]
Next, we need to calculate the height of the cylinder. The height of the cylinder is equal to the diameter of the sphere, which is \( 2r \).
The formula for the volume \( V_c \) of a cylinder is given by:
\[ V_c = \pi r^2 h \]
Substituting \( h = 2r \):
\[ V_c = \pi r^2 (2r) = 2\pi r^3 \]
Now, substituting \( r^3 = \frac{36}{\pi} \) into the volume formula of the cylinder:
\[ V_c = 2\pi \left(\frac{36}{\pi}\right) \]
Now simplify:
\[ V_c = 2 \times 36 = 72 , \text{m}^3 \]
Thus, the volume of the cylinder is:
\[ \boxed{72} , \text{m}^3 \]