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Question

A cone is placed inside a cylinder. The apex of the cone touching the center of the top circle of the cylinder is highlighted with a dot. The cone with its base is drawn in dashed lines. The base of the cone is common with the base of the cylinder.

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 the cylinder before removing the cone is 54 in.3
54
in
.
3
, what is the volume of the amount remaining?(1 point)
Responses

40 in.3
40
in.
3
40 in. cubed

36 in.3
36
in.
3
36 in. cubed

18 in.3
18
in.
3
18 in. cubed

27 in.3

Answer 1

To find the volume of the remaining portion after removing the cone from the cylinder, we first need to calculate the volume of the cone.

The volume \( V \) of a cylinder is given by the formula: \[ V_{cylinder} = \pi r^2 h \] And the volume \( V \) of a cone is given by the formula: \[ V_{cone} = \frac{1}{3} \pi r^2 h \]

Since the cylinder and cone share the same radius \( r \) and height \( h \), we know:

1. Volume of the cylinder = \( 54 , \text{in}^3 \)
2. Volume of the cone = \( \frac{1}{3} V_{cylinder} \)

Now calculating the volume of the cone: \[ V_{cone} = \frac{1}{3} \cdot 54 , \text{in}^3 = 18 , \text{in}^3 \]

Now to find the volume remaining after carving out the cone from the cylinder, we subtract the volume of the cone from the volume of the cylinder: \[ V_{remaining} = V_{cylinder} - V_{cone} \] \[ V_{remaining} = 54 , \text{in}^3 - 18 , \text{in}^3 = 36 , \text{in}^3 \]

Thus, the volume of the amount remaining after removing the cone is: \[ \boxed{36 , \text{in}^3} \]