To solve the problem, let's first understand the relationships and formulas involved for the volumes of the cylinder, sphere, and cone.
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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): \[ V_{sphere} = \frac{4}{3} \pi r^3 \]
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Volume of a Cone (V_cone): \[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Given information:
- \( V_{cylinder} = 36 , \text{cm}^3 \)
- The radius of the cylinder is \( r \).
- The height of the cylinder \( h \) is equal to the diameter of the sphere, which is \( 2r \).
Step 1: Finding the height of the cylinder
Using the volume of the cylinder formula: \[ 36 = \pi r^2 h \] Substituting \( h = 2r \): \[ 36 = \pi r^2 (2r) \] \[ 36 = 2\pi r^3 \]
Step 2: Solving for \( r^3 \)
To isolate \( r^3 \): \[ r^3 = \frac{36}{2\pi} \] \[ r^3 = \frac{18}{\pi} \]
Step 3: Finding the volume of the sphere
Now, we can use the radius \( r \) to find the volume of the sphere: \[ V_{sphere} = \frac{4}{3} \pi r^3 \] Substituting \( r^3 = \frac{18}{\pi} \): \[ V_{sphere} = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) \] \[ V_{sphere} = \frac{4 \times 18}{3} \] \[ V_{sphere} = \frac{72}{3} = 24 , \text{cm}^3 \]
Thus, the volume of the sphere is 24 cm³.
Final answer:
24 cm³