To find the volume of the amount remaining after the cone is carved out of the cylinder, we first need to calculate the volume of the cone that was removed.
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Volume of the Cylinder (V_cylinder): The volume of a cylinder is given by the formula: \[ V_{cylinder} = \pi r^2 h \] From the information provided, the volume of the cylinder is \(54 , \text{in}^3\).
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Volume of the Cone (V_cone): The volume of a cone is given by the formula: \[ V_{cone} = \frac{1}{3} \pi r^2 h \] Since the cone has the same radius and height as the cylinder, we can relate the volumes.
The volume of the cone can be calculated as: \[ V_{cone} = \frac{1}{3} V_{cylinder} = \frac{1}{3} \times 54 , \text{in}^3 = 18 , \text{in}^3 \]
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Volume Remaining: The volume remaining after removing the cone from the cylinder is: \[ V_{remaining} = V_{cylinder} - V_{cone} \] Substituting the volumes: \[ V_{remaining} = 54 , \text{in}^3 - 18 , \text{in}^3 = 36 , \text{in}^3 \]
Thus, the volume of the amount remaining is 36 in.³.
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