To find the volume of the cylinder, we first need to determine the radius and height of the cylinder based on the given information about the sphere.
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Volume of the Sphere: The volume \( V \) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere. We know the volume of the sphere is 48 m³. Setting the equation equal to 48, we have: \[ \frac{4}{3} \pi r^3 = 48 \] To find \( r^3 \), we can rearrange this equation: \[ r^3 = \frac{48 \cdot 3}{4 \pi} = \frac{144}{4 \pi} = \frac{36}{\pi} \]
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Finding the radius \( r \): Now we compute \( r \): \[ r = \left(\frac{36}{\pi}\right)^{1/3} \]
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Diameter of the Sphere: The diameter \( d \) of the sphere is given by: \[ d = 2r \]
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Volume of the Cylinder: The volume \( V_c \) of a cylinder is given by the formula: \[ V_c = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height. In this case, the height \( h \) of the cylinder is equal to the diameter of the sphere, which is \( 2r \): \[ V_c = \pi r^2 (2r) = 2\pi r^3 \]
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Substituting for \( r^3 \): We already found that: \[ r^3 = \frac{36}{\pi} \] Now we substitute this into the cylinder's volume formula: \[ V_c = 2\pi \left(\frac{36}{\pi}\right) = 2 \cdot 36 = 72 , \text{m}^3 \]
Thus, the volume of the cylinder is \( \boxed{72} , \text{m}^3 \).