To find the volume of the remaining material after the cone is carved out of the cylinder, we first need to calculate the volume of the cone and then subtract it from the volume of the cylinder.
The volume \( V \) of a cylinder is given by the formula: \[ V_{cylinder} = \pi r^2 h \]
The volume \( V \) of a cone is given by the formula: \[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Let’s denote the volume of the cylinder as \( V_{cylinder} = 54 , \text{in}^3 \).
From the relationship between the cone and cylinder (having the same radius \( r \) and height \( h \)), we can express the volume of the cylinder and the cone.
Given that the volume of the cylinder is: \[ V_{cylinder} = 54 , \text{in}^3 \]
To find the volume of the cone, we can apply the factor for the cone: \[ V_{cone} = \frac{1}{3} V_{cylinder} = \frac{1}{3} \times 54 , \text{in}^3 = 18 , \text{in}^3 \]
Now, to find the volume of the remaining material after the cone has been 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 amount remaining is 36 in.³.
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