To solve this problem, we will use the relationships between the volumes of cones, cylinders, and spheres.
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Volume of the cylinder (V_cylinder): The volume of a cylinder is given by the formula: \[ V_{cylinder} = \pi r^2 h \] where \(r\) is the radius and \(h\) is the height.
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Volume of the sphere (V_sphere): The volume of a sphere is given by the formula: \[ V_{sphere} = \frac{4}{3} \pi r^3 \]
In this case:
- The volume of the cylinder is given: \(V_{cylinder} = 36 , \text{cm}^3\).
- We need to relate the dimensions: the height of the cylinder \(h\) is equal to the sphere's diameter. Since the diameter of the sphere is \(2r\), we have \(h = 2r\).
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Substitute \(h = 2r\) into the volume of the cylinder formula: \[ V_{cylinder} = \pi r^2 (2r) = 2 \pi r^3 \] Set this equal to the volume of the cylinder: \[ 2 \pi r^3 = 36 \] Solve for \(r^3\): \[ r^3 = \frac{36}{2 \pi} = \frac{18}{\pi} \]
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Now, we can find the volume of the sphere using the derived value of \(r\): \[ V_{sphere} = \frac{4}{3} \pi r^3 \] Substitute \(r^3\) into the volume of the sphere formula: \[ V_{sphere} = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) = \frac{4 \cdot 18}{3} = \frac{72}{3} = 24 , \text{cm}^3 \]
Thus, the volume of the sphere is 24 cm³.