To solve this problem, let's start by recalling the formulas for the volumes of a cylinder, sphere, and cone:
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Volume of a cylinder (\( V_c \)): \[ V_c = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.
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Volume of a sphere (\( V_s \)): \[ V_s = \frac{4}{3} \pi r^3 \] where \( r \) is the radius.
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Volume of a cone (\( V_{co} \)): \[ V_{co} = \frac{1}{3} \pi r^2 h \]
Given that the volume of the cylinder is 36 cm³ and the height of the cylinder (\( h \)) is equal to the diameter of the sphere, we can express these relationships stepwise.
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Volume of the Cylinder: \[ V_c = \pi r^2 h = 36 , \text{cm}^3 \]
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Relating Cylinder Height and Sphere Radius: The height of the cylinder is equal to the diameter of the sphere: \[ h = 2r \] Now, we can substitute \( h \) back into the cylinder volume formula: \[ \pi r^2 (2r) = 36 \] Simplifying this: \[ 2\pi r^3 = 36 \] \[ \pi r^3 = 18 \] \[ r^3 = \frac{18}{\pi} \]
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Finding the Volume of the Sphere: Now we will now substitute \( r \) into the sphere volume formula: \[ V_s = \frac{4}{3} \pi r^3 \] Replacing \( r^3 \) with \( \frac{18}{\pi} \): \[ V_s = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) \] Simplifying this: \[ V_s = \frac{4 \times 18}{3} = \frac{72}{3} = 24 , \text{cm}^3 \]
Thus, the volume of the sphere is 24 cm³, which corresponds to option 2. 24 cm³.