To find the volume of the cylinder, we first need to know the radius of the sphere. The formula for the volume of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]
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 isolate \( r^3 \), we first multiply both sides by \( \frac{3}{4} \):
\[ \pi r^3 = 48 \times \frac{3}{4} = 36 \]
Next, divide both sides by \( \pi \):
\[ r^3 = \frac{36}{\pi} \]
Now, to find the radius \( r \):
\[ r = \sqrt[3]{\frac{36}{\pi}} \]
Next, we need to find the diameter of the sphere, which is \( 2r \).
The volume \( V_c \) of a cylinder is calculated using the formula:
\[ V_c = \pi r^2 h \]
where \( h \) (the height) is equal to the diameter of the sphere \( 2r \):
\[ V_c = \pi r^2 (2r) = 2\pi r^3 \]
Now we can substitute \( r^3 \) from our earlier calculation:
\[ V_c = 2\pi \left(\frac{36}{\pi}\right) \]
Simplifying this:
\[ V_c = 2 \times 36 = 72 , m^3 \]
Thus, the volume of the cylinder is \( \boxed{72 , m^3} \).
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