To find the volume remaining after the cone is carved out of the cylinder, we first need to calculate the volume of the cone and then subtract it from the volume of the cylinder.
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Volume of the Cylinder (V_cylinder): The volume of a cylinder is given by the formula: \[ V_{cylinder} = \pi r^2 h \] We know from the problem statement that the volume of the cylinder is 54 in³.
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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 express its volume in terms of the cylinder's volume: \[ V_{cone} = \frac{1}{3} V_{cylinder} = \frac{1}{3} \times 54 \text{ in}^3 = 18 \text{ in}^3 \]
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Volume Remaining (V_remaining): Now, to find the volume remaining after the cone is carved out, we subtract the volume of the cone from the volume of the cylinder: \[ V_{remaining} = V_{cylinder} - V_{cone} = 54 \text{ in}^3 - 18 \text{ in}^3 = 36 \text{ in}^3 \]
Thus, the volume of the amount remaining after the cone is removed from the cylinder is 36 in³.