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? (1 point) Responses sphere = cylinder + cone sphere = cylinder + cone 3 cones = 1 cylinder 3 cones = 1 cylinder 1 cylinder = 2 cones + 1 sphere 1 cylinder = 2 cones + 1 sphere sphere - cone = cylinder
Answers
GPT-4o mini
To analyze the relationship between the volumes of the sphere, cylinder, and cone, we can start by recalling their volume formulas:
1. **Sphere**: The volume \( V_s \) is given by the formula:
\[
V_s = \frac{4}{3} \pi r^3
\]
2. **Cylinder**: The volume \( V_c \) is given by the formula:
\[
V_c = \pi r^2 h
\]
3. **Cone**: The volume \( V_{cone} \) is given by the formula:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Given that the cylinder and cone share the same height \( h \), we can express the volumes in terms of \( r \) and \( h \):
- For the cylinder, substitute \( h \):
\[
V_c = \pi r^2 h
\]
- For the cone:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Next, let's evaluate how many cones fit into the volume of a cylinder:
- Since \( V_c = \pi r^2 h \) and \( V_{cone} = \frac{1}{3} \pi r^2 h \), we can see that:
\[
V_c = 3 V_{cone}
\]
This means that **3 cones fit into 1 cylinder**, which corresponds to the statement:
**3 cones = 1 cylinder**.
Now let's check the other options to clarify:
- **sphere = cylinder + cone**: This isn't correct since we haven't established that.
- **1 cylinder = 2 cones + 1 sphere**: This is unlikely from our calculations.
- **sphere - cone = cylinder**: This gives the impression of an equation we haven't explored, so it's also likely incorrect.
Therefore, the only valid relationship derived from the formulas is:
**3 cones = 1 cylinder**.
1. **Sphere**: The volume \( V_s \) is given by the formula:
\[
V_s = \frac{4}{3} \pi r^3
\]
2. **Cylinder**: The volume \( V_c \) is given by the formula:
\[
V_c = \pi r^2 h
\]
3. **Cone**: The volume \( V_{cone} \) is given by the formula:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Given that the cylinder and cone share the same height \( h \), we can express the volumes in terms of \( r \) and \( h \):
- For the cylinder, substitute \( h \):
\[
V_c = \pi r^2 h
\]
- For the cone:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Next, let's evaluate how many cones fit into the volume of a cylinder:
- Since \( V_c = \pi r^2 h \) and \( V_{cone} = \frac{1}{3} \pi r^2 h \), we can see that:
\[
V_c = 3 V_{cone}
\]
This means that **3 cones fit into 1 cylinder**, which corresponds to the statement:
**3 cones = 1 cylinder**.
Now let's check the other options to clarify:
- **sphere = cylinder + cone**: This isn't correct since we haven't established that.
- **1 cylinder = 2 cones + 1 sphere**: This is unlikely from our calculations.
- **sphere - cone = cylinder**: This gives the impression of an equation we haven't explored, so it's also likely incorrect.
Therefore, the only valid relationship derived from the formulas is:
**3 cones = 1 cylinder**.