To find the volume of the remaining amount after carving the cone out of the cylinder, we first need to determine the volume of the cone.
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Since the cone and the cylinder have the same radius \( r \) and height \( h \), we can express the volume of the cylinder and the cone in terms of \( r \) and \( h \).
Given that the volume of the cylinder is \( 54 , \text{in}^3 \),
\[ \pi r^2 h = 54 \]
Now, we find the volume of the cone:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Substituting the expression for \( \pi r^2 h \):
\[ V_{cone} = \frac{1}{3} \times 54 = 18 , \text{in}^3 \]
Now, we can find the remaining volume after removing the cone from the cylinder:
\[ \text{Remaining Volume} = \text{Volume of Cylinder} - \text{Volume of Cone} \] \[ \text{Remaining Volume} = 54 - 18 = 36 , \text{in}^3 \]
So, the volume of the amount remaining is:
\[ \text{Remaining Volume} = 36 , \text{in}^3 \]
Thus, the correct answer is 36 in.³.
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