To find the volume of the sphere given the volume of the cylinder, we'll first outline the relationships and formulas for each of the shapes.
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Volume of a Cylinder: The volume \( V_c \) of a cylinder is given by the formula: \[ V_c = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.
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Volume of a Sphere: The volume \( V_s \) of a sphere is given by the formula: \[ V_s = \frac{4}{3} \pi r^3 \]
Given that the volume of the cylinder \( V_c \) is \( 36 , \text{cm}^3 \), we can write: \[ \pi r^2 h = 36 \]
Next, we are told that the height of the cylinder is equal to the diameter of the sphere. Since the diameter of the sphere is \( 2r \), we have: \[ h = 2r \]
Substituting \( h = 2r \) into the volume formula for the cylinder, we get: \[ \pi r^2 (2r) = 36 \] This simplifies to: \[ 2\pi r^3 = 36 \] Dividing both sides by \( 2\pi \): \[ r^3 = \frac{36}{2\pi} = \frac{18}{\pi} \]
Now, we can calculate \( r \): \[ r = \left(\frac{18}{\pi}\right)^{1/3} \]
Now we need to find the volume of the sphere using the radius \( r \). Plugging \( r^3 \) back into the sphere volume formula: \[ V_s = \frac{4}{3} \pi r^3 \] Substituting \( r^3 = \frac{18}{\pi} \): \[ V_s = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) \] Simplifying this: \[ V_s = \frac{4}{3} \cdot 18 = \frac{72}{3} = 24 \]
Thus, the volume of the sphere is \( \boxed{24 , \text{cm}^3} \).