To solve this problem, let's first clarify the relationships among the volumes of cones, cylinders, and spheres.
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Volume of a cylinder (V_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 a sphere (V_sphere) is given by the formula: \[ V_{sphere} = \frac{4}{3} \pi r^3 \] where \( r \) is the radius.
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Given: The volume of the cylinder is \( 36 , \text{cm}^3 \), and we also know that the height of the cylinder is equal to the sphere's diameter.
Let \( r \) be the radius of both the cylinder and the sphere. The height \( h \) of the cylinder is equal to the diameter of the sphere, which is given by: \[ h = 2r \]
Now, substituting \( h = 2r \) into the volume formula for the cylinder, we have: \[ V_{cylinder} = \pi r^2 (2r) = 2\pi r^3 \]
We know the volume of the cylinder is \( 36 , \text{cm}^3 \): \[ 2\pi r^3 = 36 \]
To find \( r^3 \), we can solve for \( r^3 \): \[ \pi r^3 = \frac{36}{2} = 18 \] \[ r^3 = \frac{18}{\pi} \]
Now, we can calculate the volume of the sphere using the radius \( r \): \[ V_{sphere} = \frac{4}{3} \pi r^3 \] Substituting \( r^3 = \frac{18}{\pi} \): \[ V_{sphere} = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) \] \[ = \frac{4 \times 18}{3} \] \[ = \frac{72}{3} = 24 , \text{cm}^3 \]
Therefore, the volume of the sphere is 24 cm³.
The correct response is: 24 cm³.
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