To determine the volume remaining after carving a cone out of a cylinder, we need to calculate the volume of both the cylinder and the cone, and then subtract the volume of the cone from the volume of the cylinder.
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Volume of the Cylinder: The formula for the volume of a cylinder is given by: \[ V_{\text{cylinder}} = \pi r^2 h \] You mentioned that the volume of the cylinder is 54 in³.
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Volume of the Cone: The formula for the volume of a cone is given by: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Since the cone shares the same radius and height as the cylinder, we can use the same r and h.
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Volume Remaining: To find the remaining volume after the cone is removed from the cylinder, we subtract the volume of the cone from the volume of the cylinder: \[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} \]
Now substituting \( V_{\text{cone}} \): \[ V_{\text{remaining}} = V_{\text{cylinder}} - \frac{1}{3} V_{\text{cylinder}} = 54 - \frac{1}{3} \times 54 \] \[ V_{\text{remaining}} = 54 - 18 = 36 \text{ in}^3 \]
Thus, the volume remaining after carving the cone out of the cylinder is 36 in³.