To find the volume of the cylinder, we first need to understand the relationship between the volume of the sphere and its dimensions.
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Volume of the Sphere: The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Given that the volume of the sphere is \( 36 , \text{m}^3 \), we can set up the equation:
\[ \frac{4}{3} \pi r^3 = 36 \]
To solve for the radius \( r \), we rearrange the equation:
\[ r^3 = \frac{36 \times 3}{4 \pi} \]
Simplifying this:
\[ r^3 = \frac{108}{4 \pi} = \frac{27}{\pi} \]
Taking the cube root:
\[ r = \left( \frac{27}{\pi} \right)^{\frac{1}{3}} \]
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Finding the Height of the Cylinder: The height \( h \) of the cylinder is equal to the diameter of the sphere. The diameter \( d \) is given by:
\[ d = 2r \]
Thus, the height of the cylinder is:
\[ h = 2r = 2 \left( \frac{27}{\pi} \right)^{\frac{1}{3}} \]
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Volume of the Cylinder: The formula for the volume \( V_c \) of a cylinder is:
\[ V_c = \pi r^2 h \]
Substituting \( h = 2r \):
\[ V_c = \pi r^2 (2r) = 2 \pi r^3 \]
We already have \( r^3 \):
\[ V_c = 2 \pi \left( \frac{27}{\pi} \right) = 2 \times 27 = 54 , \text{m}^3 \]
Thus, the volume of the cylinder is:
\[ \boxed{54 , \text{m}^3} \]