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)
sphere cylinder + cone
3 cones 1 cylinder
1 cylinder 2 cones + 1 sphere
sphere-cone = cylinder
(1 point)
sphere cylinder + cone
3 cones 1 cylinder
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, let's start by identifying their volume formulas, assuming they have a radius \( r \) and a height \( h \).
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, given that the cylinder and cone share the same height \( h \), we can express the volume of the cylinder in terms of the volume of the cone:
Since \( V_{\text{cone}} = \frac{1}{3} \pi r^2 h \), the volume of one cone is one-third of the volume of the cylinder (which uses the same base area \( \pi r^2 \) and height \( h \)):
\[
V_{\text{cylinder}} = 3 V_{\text{cone}}.
\]
Next, let's find a relationship involving all three volumes.
To compare the spheres and cones,
\[
V_{\text{cylinder}} = 3 V_{\text{cone}} \implies V_{\text{cylinder}} = 3 \left( \frac{1}{3} \pi r^2 h \right) = \pi r^2 h.
\]
To express a possible relationship between the volume of the sphere and the volumes of the cylinder and cone, we can substitute for \( h \) in terms of the volume of the sphere.
Using \( h \) directly from the cone's formula,
\[
V_{\text{cylinder}} = \pi r^2 h \quad \text{and} \quad V_{\text{cone}} = \frac{1}{3} \pi r^2 h \implies h = \frac{3 V_{\text{cone}}}{\pi r^2}.
\]
However, to analyze the options given, the important realization is:
- The sphere has a volume \( \frac{4}{3} \pi r^3 \).
- The cone's volume is one-third of the cylinder's volume, thus the cylinder is three times the volume of a cone.
This yields:
\[
V_{\text{sphere}} < V_{\text{cylinder}} + V_{\text{cone}}.
\]
Testing provided options:
1. **sphere = cylinder + cone**: FALSE (as shown).
2. **3 cones = 1 cylinder**: TRUE (as already established).
3. **1 cylinder = 2 cones + 1 sphere**: Likely FALSE (depends on the exact relationship shown).
4. **sphere - cone = cylinder**: NOT TRUE since the equation does not hold as \( V_{\text{combination}} \).
Given all of the above analysis, the correct statement among the provided options is:
**3 cones = 1 cylinder.**
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, given that the cylinder and cone share the same height \( h \), we can express the volume of the cylinder in terms of the volume of the cone:
Since \( V_{\text{cone}} = \frac{1}{3} \pi r^2 h \), the volume of one cone is one-third of the volume of the cylinder (which uses the same base area \( \pi r^2 \) and height \( h \)):
\[
V_{\text{cylinder}} = 3 V_{\text{cone}}.
\]
Next, let's find a relationship involving all three volumes.
To compare the spheres and cones,
\[
V_{\text{cylinder}} = 3 V_{\text{cone}} \implies V_{\text{cylinder}} = 3 \left( \frac{1}{3} \pi r^2 h \right) = \pi r^2 h.
\]
To express a possible relationship between the volume of the sphere and the volumes of the cylinder and cone, we can substitute for \( h \) in terms of the volume of the sphere.
Using \( h \) directly from the cone's formula,
\[
V_{\text{cylinder}} = \pi r^2 h \quad \text{and} \quad V_{\text{cone}} = \frac{1}{3} \pi r^2 h \implies h = \frac{3 V_{\text{cone}}}{\pi r^2}.
\]
However, to analyze the options given, the important realization is:
- The sphere has a volume \( \frac{4}{3} \pi r^3 \).
- The cone's volume is one-third of the cylinder's volume, thus the cylinder is three times the volume of a cone.
This yields:
\[
V_{\text{sphere}} < V_{\text{cylinder}} + V_{\text{cone}}.
\]
Testing provided options:
1. **sphere = cylinder + cone**: FALSE (as shown).
2. **3 cones = 1 cylinder**: TRUE (as already established).
3. **1 cylinder = 2 cones + 1 sphere**: Likely FALSE (depends on the exact relationship shown).
4. **sphere - cone = cylinder**: NOT TRUE since the equation does not hold as \( V_{\text{combination}} \).
Given all of the above analysis, the correct statement among the provided options is:
**3 cones = 1 cylinder.**