To solve the problem, we first need to understand the relationships between the volumes of the cylinder, sphere, and cone.
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Volume of a Cylinder: The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.
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Volume of a Sphere: The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
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Given Data: The volume of the cylinder is \( 36 , \text{cm}^3 \) and the height of the cylinder \( h \) is equal to the sphere's diameter, which is \( 2r \).
Therefore, we can write the volume of the cylinder as:
\[ V = \pi r^2 (2r) = 2\pi r^3 \]
Setting this equal to the given volume of the cylinder:
\[ 2\pi r^3 = 36 \]
Dividing both sides by \( 2\pi \):
\[ r^3 = \frac{36}{2\pi} = \frac{18}{\pi} \]
Now, to find the volume of the sphere with this radius, we substitute \( r^3 \) back into the formula for the sphere's volume:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Thus,
\[ V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{18}{\pi} \right) \]
Simplifying this gives:
\[ V_{\text{sphere}} = \frac{4 \times 18}{3} = \frac{72}{3} = 24 , \text{cm}^3 \]
So, the volume of the sphere is 24 cm³.