Question
Question Suppose a sphere, cylinder, and cone all share the same radius, and the cylinder and cone also share the same height. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?
Answers
GPT-4o mini
To analyze the relationships between the volumes of a sphere, cylinder, and cone that share the same radius \( r \), and where the cylinder and cone share the same height \( h \), we can use the volume formulas for each shape.
1. **Volume of the Sphere**:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
2. **Volume of the Cylinder**:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
3. **Volume of the Cone**:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
Now, let's compare these volumes:
- **Ratio of the Cylinder to the Cone**:
Since the volume of the cone is one-third that of the cylinder:
\[
V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}}
\]
This means the volume of the cone is always one-third the volume of the cylinder when they share the same radius and height.
- **Comparing Sphere and Cylinder**:
The relationship between the sphere and cylinder is not as straightforward without specific values for \( r \) and \( h \). However, if we consider the volumes individually:
If we set \( h = 2r \) (a common height for comparison), we can examine:
\[
V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3
\]
Thus, under this assumption:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
\[
V_{\text{cylinder}} = 2\pi r^3
\]
\[
V_{\text{cone}} = \frac{1}{3} (2\pi r^3) = \frac{2}{3} \pi r^3
\]
In conclusion, the relationships can be summarized as:
- The volume of the cone is always one-third the volume of the cylinder with the same radius and height.
- The volume of the sphere is less than that of the cylinder when \( h \geq \frac{4}{3} r \), but the exact relationship can change based on the chosen height in relation to the radius.
In general, you can state:
- \( V_{\text{cone}} < V_{\text{sphere}} < V_{\text{cylinder}} \) when certain conditions on \( h \) are met, particularly that \( h \) is sufficiently greater than \( r \).
1. **Volume of the Sphere**:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
2. **Volume of the Cylinder**:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
3. **Volume of the Cone**:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
Now, let's compare these volumes:
- **Ratio of the Cylinder to the Cone**:
Since the volume of the cone is one-third that of the cylinder:
\[
V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}}
\]
This means the volume of the cone is always one-third the volume of the cylinder when they share the same radius and height.
- **Comparing Sphere and Cylinder**:
The relationship between the sphere and cylinder is not as straightforward without specific values for \( r \) and \( h \). However, if we consider the volumes individually:
If we set \( h = 2r \) (a common height for comparison), we can examine:
\[
V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3
\]
Thus, under this assumption:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
\[
V_{\text{cylinder}} = 2\pi r^3
\]
\[
V_{\text{cone}} = \frac{1}{3} (2\pi r^3) = \frac{2}{3} \pi r^3
\]
In conclusion, the relationships can be summarized as:
- The volume of the cone is always one-third the volume of the cylinder with the same radius and height.
- The volume of the sphere is less than that of the cylinder when \( h \geq \frac{4}{3} r \), but the exact relationship can change based on the chosen height in relation to the radius.
In general, you can state:
- \( V_{\text{cone}} < V_{\text{sphere}} < V_{\text{cylinder}} \) when certain conditions on \( h \) are met, particularly that \( h \) is sufficiently greater than \( r \).