To solve the problem, we need to use the formulas for the volume of a sphere and the volume of a cylinder.
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Volume of a sphere: \[ V = \frac{4}{3} \pi r^3 \] Given the volume of the sphere \( V = 48 , m^3 \), we can set up the equation: \[ \frac{4}{3} \pi r^3 = 48 \] To find \( r \), we first isolate \( r^3 \): \[ r^3 = \frac{48 \times 3}{4\pi} = \frac{144}{4\pi} = \frac{36}{\pi} \]
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Calculating the radius: \[ r = \left(\frac{36}{\pi}\right)^{\frac{1}{3}} \]
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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: \[ V = \pi r^2 h \] Here, the height \( h \) of the cylinder is equal to the diameter of the sphere: \[ h = 2r \] Now we can substitute \( h \) into the cylinder volume formula: \[ V = \pi r^2 (2r) = 2\pi r^3 \]
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Substituting for \( r^3 \): From our earlier calculation, we found that \( r^3 = \frac{36}{\pi} \): \[ V = 2\pi \left(\frac{36}{\pi}\right) = 2 \times 36 = 72 , m^3 \]
Thus, the volume of the cylinder is 72 m³.