To find the volume of the remaining material after the cone is carved out of the cylinder, we need to first determine 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 the cylinder is given as 54 in³.
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Volume of the Cone (V_cone): The formula for the volume of a cone is: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] In this case, since the cone and cylinder share the same radius (r) and height (h), we can determine the radius and height using the volume of the cylinder.
The volume of the cylinder is given by: \[ V_{\text{cylinder}} = \pi r^2 h \] Setting this equal to 54 in³: \[ \pi r^2 h = 54 \]
The volume of the cone is: \[ V_{\text{cone}} = \frac{1}{3} \cdot (54) = 18 \text{ in}^3 \]
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Volume Remaining: To find the volume of the material remaining after carving out the cone, we subtract the volume of the cone from the volume of the cylinder: \[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} = 54 - 18 = 36 \text{ in}^3 \]
Therefore, the volume of the amount remaining is 36 in.³.