The volume \( V \) of a cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r^2 h \]
The volume \( V \) of a cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Since the cone and the cylinder share the same radius \( r \) and height \( h \), we can calculate the volume of the cone given the volume of the cylinder.
From the problem, we know:
\[ V_{\text{cylinder}} = 54 , \text{in}^3 \]
Now, let's find the volume of the cone:
\[ V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} = \frac{1}{3} \times 54 , \text{in}^3 = 18 , \text{in}^3 \]
To find the volume of the amount remaining after carving out the cone from the cylinder, we subtract the volume of the cone from the volume of the cylinder:
\[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} = 54 , \text{in}^3 - 18 , \text{in}^3 = 36 , \text{in}^3 \]
Therefore, the volume of the amount remaining after the cone is carved out of the cylinder is:
36 in.³