To find the volume of the remaining material after a cone is carved out of a cylinder, we first need to calculate the volume of both the cylinder and the cone.
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Volume of the Cylinder: The formula for the volume of a cylinder is given by: \[ V_{cylinder} = \pi r^2 h \] You've stated that the volume of the cylinder is \(54 , \text{in}^3\).
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Volume of the Cone: The formula for the volume of a cone is: \[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Since the cone and cylinder share the same radius \(r\) and height \(h\), we can express the volume of the cone in terms of the volume of the cylinder.
Since we know the volume of the cylinder, we can write: \[ V_{cone} = \frac{1}{3} V_{cylinder} = \frac{1}{3} \times 54 , \text{in}^3 = 18 , \text{in}^3 \]
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Volume of the Remaining Material: To find the volume of the remaining material after the cone is removed from the cylinder, we subtract the volume of the cone from the volume of the cylinder: \[ V_{remaining} = V_{cylinder} - V_{cone} = 54 , \text{in}^3 - 18 , \text{in}^3 = 36 , \text{in}^3 \]
Thus, the volume of the material remaining after the cone is carved out of the cylinder is \(36 , \text{in}^3\).
Answer: 36 in³
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