To find the volume of the remaining portion after the cone is carved out of the cylinder, we first need to calculate the volume of the cone.
The volume of a cylinder can be calculated using the formula: \[ V_{\text{cylinder}} = \pi r^2 h \]
And the volume of a cone can be calculated using the formula: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Given that the volume of the cylinder before removing the cone is \(54 , \text{in}^3\), we can use that information to find the volume of the cone and then subtract it from the volume of the cylinder to find the remaining volume.
Since both the cone and the cylinder have the same radius \(r\) and height \(h\), the volume of the cone can be calculated as follows:
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If the volume of the cylinder is \(54 , \text{in}^3\), then: \[ V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} = \frac{1}{3} \times 54 = 18 , \text{in}^3 \]
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Now, we find the remaining volume after the cone is removed: \[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} = 54 , \text{in}^3 - 18 , \text{in}^3 = 36 , \text{in}^3 \]
Thus, the volume of the amount remaining is \(36 , \text{in}^3\).
The correct response is: 36 in.3 or 36 in. cubed.
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