To find the volume of the sphere with the same radius as the cylinder and a height of the cylinder that equals the diameter of the sphere, we can start by recalling the formulas for the volumes of each shape:
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Volume of a Cylinder (V_cylinder): \[ V_cylinder = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.
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Volume of a Sphere (V_sphere): \[ V_sphere = \frac{4}{3} \pi r^3 \]
Given the volume of the cylinder is \( 36 , \text{cm}^3 \), we can set up an equation. Since the height of the cylinder is equal to the diameter of the sphere, and the diameter of the sphere is \( 2r \), we have \( h = 2r \).
Substituting this into the volume formula for the cylinder gives: \[ V_cylinder = \pi r^2 (2r) = 2\pi r^3 \]
Setting this equal to \( 36 , \text{cm}^3 \): \[ 2\pi r^3 = 36 \]
Now, we can solve for \( r^3 \): \[ \pi r^3 = 18 \] \[ r^3 = \frac{18}{\pi} \]
Now let’s find the volume of the sphere using the radius \( r \): \[ V_sphere = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left( \frac{18}{\pi} \right) \]
Breaking this down: \[ V_sphere = \frac{4}{3} \cdot 18 = 24 , \text{cm}^3 \]
Thus, the volume of the sphere is 24 cm³.
The correct answer is 24 cm³.
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