To find the volume of the cylinder given that the radius is the same as the sphere's and the height is equal to the sphere's diameter, we need to follow these steps:
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Volume of the Sphere: The formula for the volume \( V \) of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \] We know the volume of the sphere is 48 m³. Therefore, \[ \frac{4}{3} \pi r^3 = 48 \]
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Solving for the Radius: To isolate \( r^3 \): \[ r^3 = \frac{48 \cdot 3}{4 \pi} = \frac{144}{4 \pi} = \frac{36}{\pi} \] Thus, the radius \( r \) can be found by taking the cube root: \[ r = \left( \frac{36}{\pi} \right)^{1/3} \]
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Calculating the Diameter of the Sphere: The diameter \( d \) of the sphere is given by: \[ d = 2r \] Hence, we can express \( d \) as: \[ d = 2 \left( \frac{36}{\pi} \right)^{1/3} \]
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Volume of the Cylinder: The volume \( V_c \) of a cylinder is given by: \[ V_c = \pi r^2 h \] where \( h \) is the height of the cylinder. Since the height \( h \) is equal to the diameter of the sphere: \[ h = d = 2r \] Substituting this in, we have: \[ V_c = \pi r^2 (2r) = 2 \pi r^3 \]
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Substituting the Value of \( r^3 \): From our previous calculation, we have \( r^3 = \frac{36}{\pi} \). Now substituting into the volume formula: \[ V_c = 2 \pi \left( \frac{36}{\pi} \right) = 2 \cdot 36 = 72 \]
Thus, the volume of the cylinder is \( \boxed{72} \) m³.