To solve this problem, we will first outline the relevant formulas and relationships among the volumes of a cylinder, sphere, and cone.
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Volume of a Cylinder: The formula for the volume of a cylinder is given by:
\[ V_{cylinder} = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.
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Volume of a Sphere: The formula for the volume of a sphere is given by:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius.
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Relationship: In the problem, we have:
- The volume of the cylinder \( V_{cylinder} = 36 \) cm³.
- The radius of the sphere is the same as the radius of the cylinder \( r \).
- The height of the cylinder is equal to the diameter of the sphere, hence \( h = 2r \).
Using the given volume of the cylinder, we can find the radius.
\[ 36 = \pi r^2 h \]
Substituting \( h = 2r \):
\[ 36 = \pi r^2 (2r) = 2\pi r^3 \]
Now, solving for \( r^3 \):
\[ r^3 = \frac{36}{2\pi} = \frac{18}{\pi} \]
Next, we substitute \( r^3 \) into the volume formula for the sphere:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]
Substituting our expression for \( r^3 \):
\[ V_{sphere} = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) = \frac{4}{3} \cdot 18 = 24 \]
Thus, the volume of the sphere is:
\[ \boxed{24 \text{ cm}^3} \]
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