To solve this problem, we first need to find the volume of the cone that was carved out of the cylinder, and then we can find the volume of the remaining material by subtracting the volume of the cone from the volume of the cylinder.
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Volume of the Cylinder: The volume \( V \) of a cylinder is given by the formula: \[ V = \pi r^2 h \] It is stated that the volume of the cylinder is 54 in³.
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Volume of the Cone: The volume \( V_c \) of a cone is given by the formula: \[ V_c = \frac{1}{3} \pi r^2 h \] Since the cone and cylinder share the same radius \( r \) and height \( h \), we can find the volume of the cone using the volume of the cylinder.
The volume of the cone can be expressed in terms of the volume of the cylinder: \[ V_c = \frac{1}{3} \times \text{Volume of the cylinder} \] Therefore: \[ V_c = \frac{1}{3} \times 54 = 18 \text{ in}^3 \]
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Volume of the Remaining Material: To find the volume of the material remaining after the cone has been carved out, we subtract the volume of the cone from the volume of the cylinder: \[ \text{Volume remaining} = \text{Volume of the cylinder} - \text{Volume of the cone} \] \[ \text{Volume remaining} = 54 \text{ in}^3 - 18 \text{ in}^3 = 36 \text{ in}^3 \]
Therefore, the volume of the amount remaining after the cone has been removed is 36 in³.
The correct answer is 3. 36 in³.
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