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.
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Volume of the sphere: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
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Volume of the cylinder: \[ V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]
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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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