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
, what is the volume of the amount remaining?(1 point)
Responses

40 in.3
40 in. cubed

27 in.3
27 in. cubed

18 in.3
18 in. cubed

36 in.3

Answer 1

To find the volume of the remaining material after carving a cone out of a cylinder, we first need to calculate the volume of both the cylinder and the cone, and then subtract the volume of the cone from the volume of the cylinder.

1. **Volume of the cylinder:**
The volume \( V_c \) of a cylinder is given by the formula:
\[
V_c = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.

We are given that the volume of the cylinder is 54 in³.

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

Since the cone and cylinder have the same radius and height, the volume of the cone can be calculated as:
\[
V_{cone} = \frac{1}{3} V_c = \frac{1}{3} \times 54 = 18 \text{ in}^3
\]

3. **Volume of the remaining material:**
Now, we can find the volume of the remaining material:
\[
V_{remaining} = V_c - V_{cone} = 54 - 18 = 36 \text{ in}^3
\]

Therefore, the volume of the amount remaining is **36 in.³**.