To find the volume of the original cylinder, we first need to understand the relationship between the cylinder volume and the cone volume.
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 of the cylinder.
The volume \( V \) of a cone is given by the formula:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Given that the cone and the cylinder share the same radius and height, the volume relationship can be simplified. The volume of the remaining material after carving out the cone from the cylinder is:
\[ V_{remaining} = V_{cylinder} - V_{cone} \]
Substituting the formulas for the volumes, we have:
\[ V_{remaining} = \pi r^2 h - \frac{1}{3} \pi r^2 h \]
Factoring out \( \pi r^2 h \):
\[ V_{remaining} = \pi r^2 h \left(1 - \frac{1}{3}\right) = \pi r^2 h \left(\frac{2}{3}\right) \]
We know that the remaining volume is 144 in³:
\[ \frac{2}{3} \pi r^2 h = 144 \]
To find the volume of the cylinder, we can express it in terms of \( V_{remaining} \):
\[ V_{cylinder} = V_{remaining} + V_{cone} = 144 + \frac{1}{3} \cdot V_{cylinder} \]
Let \( V_{cylinder} = V \). Thus:
\[ V = 144 + \frac{1}{3} V \]
Now, multiplying through by 3 to eliminate the fraction:
\[ 3V = 432 + V \]
Rearranging gives:
\[ 3V - V = 432 \]
\[ 2V = 432 \]
Now, dividing both sides by 2:
\[ V = 216 \]
Thus, the volume of the original cylinder is 216 in³ (216 in. cubed).
The correct response is: 216 in.3 or 216 in. cubed.