To solve the problem, we first need to find the volume of the cone that has been carved out of the cylinder.
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Volume of the Cylinder (V_c):
The volume of a cylinder is given by the formula: \[ V_c = \pi r^2 h \] According to the problem, the volume of the cylinder before removing the cone is \(54 , \text{in}^3\). -
Volume of the Cone (V_cone):
The volume of a cone is given by the formula: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Since the cone has the same radius and height as the cylinder, the volume of the cone will be: \[ V_{\text{cone}} = \frac{1}{3} V_c = \frac{1}{3} \times 54 , \text{in}^3 = 18 , \text{in}^3 \] -
Volume Remaining After Carving the Cone:
To find the volume remaining in the cylinder after the cone has been removed, we subtract the volume of the cone from the volume of the cylinder: \[ V_{\text{remaining}} = V_c - V_{\text{cone}} = 54 , \text{in}^3 - 18 , \text{in}^3 = 36 , \text{in}^3 \]
Thus, the volume of the amount remaining after the cone has been removed is 36 in.³.