To find the volume of the amount remaining after the cone has been removed from the cylinder, we first need to determine the volume of the cone and then subtract that from the volume of the cylinder.
-
Volume of the Cylinder:
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
We know the volume of the cylinder is 54 in³. -
Volume of the Cone:
The formula for the volume \( V \) of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]
Since the cone has the same radius and height as the cylinder, we'll use the same \( r \) and \( h \). -
Finding the Volume of the Cone:
From the volume of the cylinder, we can express \( \pi r^2 h \) as:
\[ \pi r^2 h = 54 \]
Therefore, the volume of the cone is:
\[ V_{\text{cone}} = \frac{1}{3} \cdot 54 = 18 \text{ in}^3 \] -
Volume of the Remaining Solid:
To find the volume of the remaining solid after removing the cone from the cylinder, we subtract the cone's volume from the cylinder's volume:
\[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} = 54 \text{ in}^3 - 18 \text{ in}^3 = 36 \text{ in}^3 \]
Therefore, the volume of the amount remaining is 36 in.³.