To find the volume of the original cylinder, we first need to calculate the volume of the cone that is carved out of it.
The formula for the volume of a cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
The formula for the volume of a cylinder is:
\[ V_{\text{cylinder}} = \pi r^2 h \]
Given that the cone and cylinder have the same radius \( r \) and height \( h \), we can express the volume of the cylinder in terms of the volume of the cone.
If the volume remaining after carving the cone out of the cylinder is 144 in³, we can set up an equation:
\[ V_{\text{cylinder}} - V_{\text{cone}} = 144 \]
Substituting the formulas for the volumes, we get:
\[ \pi r^2 h - \frac{1}{3} \pi r^2 h = 144 \]
Factoring out \( \pi r^2 h \):
\[ \left(1 - \frac{1}{3}\right) \pi r^2 h = 144 \]
\[ \frac{2}{3} \pi r^2 h = 144 \]
To find the volume of the cylinder, we multiply both sides of the equation by \( \frac{3}{2} \):
\[ \pi r^2 h = 144 \times \frac{3}{2} = 216 \]
Thus, the volume of the original cylinder is:
216 in³
So the correct response is:
216 in.3
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