Question
Suppose+a+sphere%2C+cylinder%2C+and+cone+all+share+the+same+radius+and+the+cylinder+and+cone+also+share+the+same+height%2C+which+is+twice+the+radius.+Which+of+the+following+is+true+about+the+relationship+between+the+volume+formulas+for+the+sphere%2C+cylinder%2C+and+cone%3F(1+point)+Responses+3+cones+%3D+1+cylinder+3+cones+%3D+1+cylinder+sphere+-+cone+%3D+cylinder+sphere+-+cone+%3D+cylinder+sphere+%3D+cylinder+%2B+cone+sphere+%3D+cylinder+%2B+cone+1+cylinder+%3D+2+cones+%2B+1+sphere
Answers
Answered by
GPT-4o mini
To analyze the relationship between the volumes of a sphere, cylinder, and cone sharing the same radius \( r \) and with the height \( h \) of the cylinder and cone equal to \( 2r \), we can start by writing down 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**:
\[
V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3
\]
3. **Volume of the cone**:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
Now we need to compare these volumes.
- **Relationship between the volumes**:
- \( V_{\text{cylinder}} = 2\pi r^3 \)
- \( V_{\text{cone}} = \frac{2}{3} \pi r^3 \)
Let’s find out how many cones are equal to one cylinder by dividing the volume of the cylinder by the volume of the cone:
\[
\text{Number of cones in a cylinder} = \frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{2\pi r^3}{\frac{2}{3}\pi r^3} = 3
\]
So, we find that:
\[
1 \text{ cylinder} = 3 \text{ cones}
\]
Next, let’s confirm the relationship with the sphere's volume:
- To see how the sphere compares to the cylinder:
\[
\text{Sphere volume} = \frac{4}{3} \pi r^3
\]
Thus, comparing it to the cylinder:
\[
\text{If we want to express the sphere’s volume in terms of the cylinder’s volume}:
\frac{4/3 \pi r^3}{2 \pi r^3} = \frac{2}{3}
\]
This indicates that the volume of the sphere is \(\frac{2}{3}\) of the volume of the cylinder.
Now we can summarize the relationships:
- **1 cylinder = 3 cones**
- **1 cylinder = 1 sphere + 2 cones** (since \(1 \text{ cylinder} (2\pi r^3) = 1 \text{ sphere} (\frac{4}{3} \pi r^3) + 2 \text{ cones} (\frac{2}{3} \pi r^3)\))
The final relationship can be stated as:
\[
\text{cylinder} = \text{sphere} + 2 \text{ cones}
\]
Thus, the correct conclusion based on your question is:
**1 cylinder = 2 cones + 1 sphere**.
1. **Volume of the sphere**:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
2. **Volume of the cylinder**:
\[
V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3
\]
3. **Volume of the cone**:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
Now we need to compare these volumes.
- **Relationship between the volumes**:
- \( V_{\text{cylinder}} = 2\pi r^3 \)
- \( V_{\text{cone}} = \frac{2}{3} \pi r^3 \)
Let’s find out how many cones are equal to one cylinder by dividing the volume of the cylinder by the volume of the cone:
\[
\text{Number of cones in a cylinder} = \frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{2\pi r^3}{\frac{2}{3}\pi r^3} = 3
\]
So, we find that:
\[
1 \text{ cylinder} = 3 \text{ cones}
\]
Next, let’s confirm the relationship with the sphere's volume:
- To see how the sphere compares to the cylinder:
\[
\text{Sphere volume} = \frac{4}{3} \pi r^3
\]
Thus, comparing it to the cylinder:
\[
\text{If we want to express the sphere’s volume in terms of the cylinder’s volume}:
\frac{4/3 \pi r^3}{2 \pi r^3} = \frac{2}{3}
\]
This indicates that the volume of the sphere is \(\frac{2}{3}\) of the volume of the cylinder.
Now we can summarize the relationships:
- **1 cylinder = 3 cones**
- **1 cylinder = 1 sphere + 2 cones** (since \(1 \text{ cylinder} (2\pi r^3) = 1 \text{ sphere} (\frac{4}{3} \pi r^3) + 2 \text{ cones} (\frac{2}{3} \pi r^3)\))
The final relationship can be stated as:
\[
\text{cylinder} = \text{sphere} + 2 \text{ cones}
\]
Thus, the correct conclusion based on your question is:
**1 cylinder = 2 cones + 1 sphere**.
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