To solve this problem, we need to first find the radius of the sphere and then use that radius to calculate the volume of the cylinder.
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Volume of the Sphere: The formula for the volume of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \] We are given that the volume \( V \) is 48 m³: \[ \frac{4}{3} \pi r^3 = 48 \]
To solve for \( r \), rearranging gives: \[ r^3 = \frac{48 \cdot 3}{4 \pi} = \frac{144}{4 \pi} = \frac{36}{\pi} \] Now let's calculate \( r \): \[ r = \left( \frac{36}{\pi} \right)^{1/3} \]
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Diameter of the Sphere: The diameter \( d \) of the sphere is: \[ d = 2r \]
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Volume of the Cylinder: The formula for the volume of a cylinder is: \[ V = \pi r^2 h \] Where \( h \) is the height. From our previous calculation, the height of the cylinder is equal to the diameter of the sphere, so: \[ h = d = 2r \] Therefore, the volume of the cylinder becomes: \[ V = \pi r^2 (2r) = 2 \pi r^3 \]
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Substituting the Volume of the Sphere: We know from the first calculation that \( \frac{4}{3} \pi r^3 = 48 \), which means: \[ r^3 = \frac{36}{\pi} \] Now substituting for \( r^3 \) in the cylinder's volume: \[ V = 2 \pi r^3 = 2 \pi \left( \frac{36}{\pi} \right) \]
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Calculating the Volume of the Cylinder: \[ V = 2 \cdot 36 = 72 , \text{m}^3 \]
Thus, the volume of the cylinder is \( \boxed{72 , \text{m}^3} \).