To find the volume of the remaining material after carving out the cone from the cylinder, we first need to determine the volume of the cone.
-
Volume of the Cylinder: The volume of the cylinder is already given as 54 in³.
-
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 and the cylinder have the same radius \( r \) and height \( h \), we need to express the volume of the cone in terms of the volume of the cylinder.
The formula for the volume of the cylinder is: \[ V_{\text{cylinder}} = \pi r^2 h \]
-
Relating the Volumes: From the given volume of the cylinder: \[ 54 = \pi r^2 h \]
Therefore, \( \pi r^2 h = 54 \).
Now substituting into the cone volume formula: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} (54) = 18 \text{ in}^3 \]
-
Volume of the Remaining Material: The volume of the remaining material after carving out the cone is: \[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} = 54 \text{ in}^3 - 18 \text{ in}^3 = 36 \text{ in}^3 \]
Thus, the volume of the amount remaining is 36 in³.