To find the volume of the amount remaining after the cone is carved out of the cylinder, we first need to know the volumes of both the cylinder and the cone.
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Volume of the Cylinder (V_cylinder):
The formula for the volume of a cylinder is: \[ V_{cylinder} = \pi r^2 h \] Given that the volume of the cylinder is 54 in³. -
Volume of the Cone (V_cone):
The formula for the volume of a cone is: \[ V_{cone} = \frac{1}{3} \pi r^2 h \] Since the problem states that the cone has the same radius and height as the cylinder, the volume of the cone can be expressed in terms of the cylinder's volume.
Since the volume of the cone is one third of the cylinder's volume, we can calculate that: \[ V_{cone} = \frac{1}{3} V_{cylinder} = \frac{1}{3} \times 54 , \text{in}^3 = 18 , \text{in}^3 \]
- Volume Remaining:
To find the volume of the remaining part of the cylinder after the cone has been removed, we subtract the volume of the cone from the volume of the cylinder: \[ \text{Remaining Volume} = V_{cylinder} - V_{cone} = 54 , \text{in}^3 - 18 , \text{in}^3 = 36 , \text{in}^3 \]
The answer is:
36 in.³ (or 36 in. cubed)