A sculptor created a design by carving a cone out of a cylinder The cone and cylinder share the same radius and height If the volume of cylinder before removing the cone is 54in .^ 3 what is the volume of the amount remaining? (1 point)
27i * n_{i} ^ 3
40in .^ 3
36in .^ 3
18 ln^ 3
To find the volume of the amount remaining after the cone is carved out of the cylinder, we first need to find the volume of the cone and then subtract it from the volume of the cylinder.
The volume \( V \) of a cone can be calculated using the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Since the cone and the cylinder have the same radius and height, we can find the volume of the cone using this formula with the values of radius and height that correspond to the cylinder.
But we don't have those specific values; we do have the total volume \( V_{\text{cylinder}} \) of the cylinder, which is:
\[ V_{\text{cylinder}} = 54 \text{ in}^3 \]
The volume \( V \) of a cylinder is calculated by the formula:
\[ V_{\text{cylinder}} = \pi r^2 h \]
Given that the cone is perfectly inscribed within the cylinder, its volume is exactly 1/3 of the volume of the cylinder, because the height and base area are the same and the cone volume formula includes the 1/3 factor.
Therefore, the volume of the cone \( V_{\text{cone}} \) is:
\[ V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} = \frac{1}{3} \times 54 \text{ in}^3 \]
\[ V_{\text{cone}} = 18 \text{ in}^3 \]
Now, we subtract the volume of the cone from the volume of the cylinder to get the remaining volume:
\[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} \]
\[ V_{\text{remaining}} = 54 \text{ in}^3 - 18 \text{ in}^3 \]
\[ V_{\text{remaining}} = 36 \text{ in}^3 \]
Thus, the volume of the amount remaining after carving the cone out of the cylinder is 36 in^3.