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
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
Answers
GPT-4o mini
To find the volume remaining after the cone is removed from the cylinder, we first need to determine the volume of the original cylinder and the volume of the cone carved out.
1. **Volume of the Cylinder:**
The formula for the volume \( V_c \) of a cylinder is given by:
\[
V_c = \pi r^2 h
\]
We are given that the volume of the cylinder is 54 in³.
2. **Volume of the Cone:**
The formula for the volume \( V_{cone} \) of a cone is given by:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Since the cone and cylinder have the same radius \( r \) and height \( h \), we can set the volume of the cone in terms of the cylinder's volume:
\[
V_{cone} = \frac{1}{3} V_c
\]
We know \( V_c = 54 \) in³, thus:
\[
V_{cone} = \frac{1}{3} \times 54 = 18 \text{ in}^3
\]
3. **Volume Remaining after Removing the Cone:**
To find the volume remaining after the cone is carved out of the cylinder, we subtract the volume of the cone from the volume of the cylinder:
\[
V_{remaining} = V_c - V_{cone}
\]
Substituting the known values:
\[
V_{remaining} = 54 - 18 = 36 \text{ in}^3
\]
Thus, the volume of the amount remaining after the cone is removed is **36 in³**.
1. **Volume of the Cylinder:**
The formula for the volume \( V_c \) of a cylinder is given by:
\[
V_c = \pi r^2 h
\]
We are given that the volume of the cylinder is 54 in³.
2. **Volume of the Cone:**
The formula for the volume \( V_{cone} \) of a cone is given by:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Since the cone and cylinder have the same radius \( r \) and height \( h \), we can set the volume of the cone in terms of the cylinder's volume:
\[
V_{cone} = \frac{1}{3} V_c
\]
We know \( V_c = 54 \) in³, thus:
\[
V_{cone} = \frac{1}{3} \times 54 = 18 \text{ in}^3
\]
3. **Volume Remaining after Removing the Cone:**
To find the volume remaining after the cone is carved out of the cylinder, we subtract the volume of the cone from the volume of the cylinder:
\[
V_{remaining} = V_c - V_{cone}
\]
Substituting the known values:
\[
V_{remaining} = 54 - 18 = 36 \text{ in}^3
\]
Thus, the volume of the amount remaining after the cone is removed is **36 in³**.