To solve this problem, let's first recall the formulas for the volumes of the three shapes involved:
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Volume of a cylinder: \[ V_{\text{cylinder}} = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.
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Volume of a sphere: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \] where \( r \) is the radius.
In this problem, we know:
- The volume of the cylinder is \( 36 , \text{cm}^3 \).
- The height of the cylinder is equal to the diameter of the sphere.
- The radius of the sphere is equal to the radius of the cylinder.
Since the diameter of the sphere is twice the radius, we have: \[ h = 2r \]
Substituting \( h = 2r \) into the formula for the volume of the cylinder, we get: \[ V_{\text{cylinder}} = \pi r^2 (2r) = 2 \pi r^3 \] Setting this equal to the given volume of the cylinder: \[ 2 \pi r^3 = 36 \] Now, we can solve for \( r^3 \): \[ \pi r^3 = 18 \quad \Rightarrow \quad r^3 = \frac{18}{\pi} \]
Now, let's calculate the volume of the sphere using the radius \( r \): \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \] Substituting \( r^3 = \frac{18}{\pi} \): \[ V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) = \frac{4 \cdot 18}{3} = \frac{72}{3} = 24 \]
Thus, the volume of the sphere is \( 24 , \text{cm}^3 \).
The correct response is: 24 cm³
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