To find the volume of the original cylinder, we first need to understand the relationship between the volumes of the cylinder and the cone.
The volume \( V \) of a cylinder is given by the formula: \[ V_{cylinder} = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.
The volume \( V \) of a cone is given by the formula: \[ V_{cone} = \frac{1}{3} \pi r^2 h \]
If the cone and cylinder share the same radius and height, the remaining volume after carving out the cone will be: \[ V_{remaining} = V_{cylinder} - V_{cone} \]
Substituting the volume formulas for the cylinder and cone, we have: \[ V_{remaining} = \pi r^2 h - \frac{1}{3} \pi r^2 h \]
Factoring out the common terms: \[ V_{remaining} = \pi r^2 h \left(1 - \frac{1}{3}\right) \] \[ V_{remaining} = \pi r^2 h \left(\frac{2}{3}\right) \] \[ V_{remaining} = \frac{2}{3} \pi r^2 h \]
We know that the remaining volume is 144 in³, so we set up the equation: \[ \frac{2}{3} \pi r^2 h = 144 \]
Now, we want to find the volume of the original cylinder: \[ V_{cylinder} = \pi r^2 h \]
Since \( V_{cylinder} = \frac{3}{2} V_{remaining} \), we can substitute \( V_{remaining} \): \[ V_{cylinder} = \frac{3}{2} \times 144 = 216 \text{ in}^3 \]
Thus, the volume of the original cylinder is: \[ \boxed{216 \text{ in}^3} \]