A cone is placed inside a cylinder. The apex of the cone touching the center of the top circle of the cylinder is highlighted with a dot. The cone with its base is drawn in dashed lines. The base of the cone is common with the base of the cylinder.

To find the volume of the remaining material after carving a cone out of a cylinder, we first 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.

1. Volume of the cylinder: The volume \( V_c \) of a cylinder is given by the formula: \[ V_c = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.

   We are given that the volume of the cylinder is 54 in³.

2. Volume of the cone: The volume \( V_{cone} \) of a cone is given by the formula: \[ V_{cone} = \frac{1}{3} \pi r^2 h \]

   Since the cone and cylinder have the same radius and height, the volume of the cone can be calculated as: \[ V_{cone} = \frac{1}{3} V_c = \frac{1}{3} \times 54 = 18 \text{ in}^3 \]

3. Volume of the remaining material: Now, we can find the volume of the remaining material: \[ V_{remaining} = V_c - V_{cone} = 54 - 18 = 36 \text{ in}^3 \]

Therefore, the volume of the amount remaining is 36 in.³.