To find the volume of the original cylinder, you first need to understand the relationship between the volumes of the cylinder and the cone. The formula for the volume of a cylinder (V_cylinder) is given by:
\[ V_{cylinder} = \pi r^2 h \]
The formula for the volume of a cone (V_cone) is given by:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Since the cone and cylinder share the same radius (r) and height (h), the volume of the remaining material after carving the cone from the cylinder is:
\[ V_{remaining} = V_{cylinder} - V_{cone} \]
Substituting the volumes into the equation gives:
\[ V_{remaining} = \pi r^2 h - \frac{1}{3} \pi r^2 h \]
Factoring out \(\pi r^2 h\):
\[ V_{remaining} = \pi r^2 h \left( 1 - \frac{1}{3} \right) = \pi r^2 h \left( \frac{3}{3} - \frac{1}{3} \right) = \frac{2}{3} \pi r^2 h \]
Now, if \(V_{remaining} = 144\) in³, then we can set up the equation:
\[ \frac{2}{3} V_{cylinder} = 144 \]
To find the volume of the cylinder, solve for \(V_{cylinder}\):
\[ V_{cylinder} = \frac{144 \cdot 3}{2} = 216 , \text{in}^3 \]
Thus, the volume of the original cylinder was 216 in³.
So the correct response is:
216 in.3 (or 216 in. cubed)