To solve this problem, we first need to understand the relationships among the volumes of a cylinder, sphere, and cone.
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Volume of a Cylinder: The volume \( V \) of a cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height. -
Volume of a Sphere: The volume \( V \) of a sphere is given by the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \] -
Relation Between Cylinder and Sphere: In this problem, we know:
- The volume of the cylinder is \( 36 , \text{cm}^3 \).
- The height of the cylinder \( h \) is equal to the diameter of the sphere, which is \( 2r \).
Using the volume formula for the cylinder, we have:
\[ V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]
Setting the volume of the cylinder equal to 36 cm³, we have:
\[ 2\pi r^3 = 36 \]
Now, solving for \( r^3 \):
\[ \pi r^3 = 18 \ r^3 = \frac{18}{\pi} \]
Now, we can use this \( r \) to find the volume of the sphere:
Substituting \( r^3 \) into the volume formula for the sphere, we get:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \ V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) \ V_{\text{sphere}} = \frac{4}{3} \cdot 18 \ V_{\text{sphere}} = \frac{72}{3} \ V_{\text{sphere}} = 24 \text{ cm}^3 \]
Thus, the volume of the sphere is 24 cm³.
Final Answer: 24 cm³