To find the volume of the remaining part after carving out the cone, we need to first determine the volume of the cylinder and the volume of the cone, then subtract the volume of the cone from the volume of the cylinder.
-
Volume of the Cylinder: The volume of a cylinder is calculated using the formula: \[ V_{\text{cylinder}} = \pi r^2 h \] You mentioned that the volume of the cylinder is \(54 \text{ in}^3\).
-
Volume of the Cone: The volume of a cone is calculated using the formula: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Since the cone shares the same radius and height as the cylinder, we can express its volume in terms of the cylinder's volume.
If we let the volume of the cylinder \(V_{\text{cylinder}} = 54 \text{ in}^3\), we can find the cone's volume as follows:
\[ V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} = \frac{1}{3} \times 54 = 18 \text{ in}^3 \]
-
Volume of the Remaining Part: Now we subtract the volume of the cone from the volume of the cylinder:
\[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} = 54 \text{ in}^3 - 18 \text{ in}^3 = 36 \text{ in}^3 \]
Therefore, the volume of the amount remaining after carving out the cone is \(36 \text{ in}^3\).
The correct response is 36 in.3 (or 36 in. cubed).