To solve this problem, we first need to understand the relationships and formulas for the volumes of a cylinder, a cone, and a sphere.
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Cylinder Volume Formula: \[ V_{cylinder} = \pi r^2 h \]
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Sphere Volume Formula: \[ V_{sphere} = \frac{4}{3} \pi r^3 \]
Given that the volume of the cylinder is 36 cm³, we can express it as: \[ \pi r^2 h = 36 \]
Next, we know that the height of the cylinder (h) is equal to the diameter of the sphere. Since the diameter of the sphere is \(2r\), we have: \[ h = 2r \]
Substituting \(h\) in the cylinder's volume formula: \[ \pi r^2 (2r) = 36 \] \[ 2\pi r^3 = 36 \]
To find the radius, we can rearrange this equation: \[ \pi r^3 = 18 \] \[ r^3 = \frac{18}{\pi} \]
Now, we can find the volume of the sphere: \[ V_{sphere} = \frac{4}{3} \pi r^3 \] Substituting the value of \(r^3\): \[ V_{sphere} = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) \] \[ = \frac{4 \cdot 18}{3} \] \[ = \frac{72}{3} \] \[ = 24 \text{ cm}^3 \]
Thus, the volume of the sphere, given the radius and the relationship to the cylinder's dimensions, is \(24 \text{ cm}^3\).
So the correct response is: 24 cm³
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