Let's denote the radius as \( r \) and the height of the cylinder and cone as \( h = 2r \).
The volume formulas for each shape are as follows:
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Sphere: \[ V_{sphere} = \frac{4}{3} \pi r^3 \]
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Cylinder: \[ V_{cylinder} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]
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Cone: \[ V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now, let's compare the volumes.
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Volume of the Cylinder: \[ V_{cylinder} = 2\pi r^3 \]
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Volume of the Cone: \[ V_{cone} = \frac{2}{3} \pi r^3 \]
Now we can check the relationship between them:
Option checking:
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cylinder = sphere – cone: \[ 2\pi r^3 \stackrel{?}{=} \frac{4}{3}\pi r^3 - \frac{2}{3}\pi r^3 \] \[ 2\pi r^3 = \frac{2}{3}\pi r^3 \quad \text{(False)} \]
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cone = sphere – cylinder: \[ \frac{2}{3}\pi r^3 \stackrel{?}{=} \frac{4}{3}\pi r^3 - 2\pi r^3 \] \[ \frac{2}{3}\pi r^3 = \frac{4}{3}\pi r^3 - \frac{6}{3}\pi r^3 \] \[ \frac{2}{3}\pi r^3 = -\frac{2}{3}\pi r^3 \quad \text{(False)} \]
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sphere = cylinder + cone: \[ \frac{4}{3}\pi r^3 \stackrel{?}{=} 2\pi r^3 + \frac{2}{3}\pi r^3 \] \[ \frac{4}{3}\pi r^3 = \left(\frac{6}{3}\pi r^3 + \frac{2}{3}\pi r^3\right) \quad \text{(True)} \]
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cone = cylinder – sphere: \[ \frac{2}{3}\pi r^3 \stackrel{?}{=} 2\pi r^3 - \frac{4}{3}\pi r^3 \] \[ \frac{2}{3}\pi r^3 = \left(\frac{6}{3}\pi r^3 - \frac{4}{3}\pi r^3\right) \quad \text{(False)} \]
Conclusion:
The true relationship is: \[ \text{sphere} = \text{cylinder} + \text{cone} \]
So the correct response is: sphere = cylinder + cone