Let's denote the radius of the sphere, cylinder, and cone as \( r \), and the height of the cylinder and cone as \( h \). According to the problem, the height \( h \) is twice the radius:
\[ h = 2r. \]
Now, we can calculate the volumes of each shape using their respective volume formulas:
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Volume of the Sphere: The formula for the volume of a sphere is given by:
\[ V_{sphere} = \frac{4}{3} \pi r^3. \]
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Volume of the Cylinder: The formula for the volume of a cylinder is:
\[ V_{cylinder} = \pi r^2 h. \]
Substituting \( h = 2r \):
\[ V_{cylinder} = \pi r^2 (2r) = 2\pi r^3. \]
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Volume of the Cone: The volume of a cone is given by the formula:
\[ V_{cone} = \frac{1}{3} \pi r^2 h. \]
Again substituting \( h = 2r \):
\[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3. \]
Now that we have all the volumes calculated, we can summarize them:
- Volume of the sphere: \( V_{sphere} = \frac{4}{3} \pi r^3 \)
- Volume of the cylinder: \( V_{cylinder} = 2 \pi r^3 \)
- Volume of the cone: \( V_{cone} = \frac{2}{3} \pi r^3 \)
Next, let's compare the volumes:
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Comparing the sphere and the cylinder:
\[ V_{sphere} = \frac{4}{3} \pi r^3, \quad V_{cylinder} = 2 \pi r^3. \] To compare them, we can express \( V_{cylinder} \) in terms of \( V_{sphere} \):
\[ V_{cylinder} = \frac{2 \cdot 3}{3} \pi r^3 = \frac{6}{3} \pi r^3. \]
This shows that:
\[ V_{cylinder} > V_{sphere}. \]
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Comparing the sphere and the cone:
\[ V_{sphere} = \frac{4}{3} \pi r^3, \quad V_{cone} = \frac{2}{3} \pi r^3. \] Clearly, since \( \frac{4}{3} > \frac{2}{3} \), we determine that:
\[ V_{sphere} > V_{cone}. \]
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Comparing the cylinder and the cone:
\[ V_{cylinder} = 2 \pi r^3, \quad V_{cone} = \frac{2}{3} \pi r^3. \] Again, comparing:
\[ 2 = \frac{6}{3} > \frac{2}{3} \Rightarrow V_{cylinder} > V_{cone}. \]
Conclusion:
We have established relationships:
- \( V_{cylinder} > V_{sphere} > V_{cone} \).
In summary, among the three volumes, the cylinder has the greatest volume, the sphere has a volume greater than the cone, and the cone has the smallest volume. Thus, the true relationship is as follows:
\[ V_{cylinder} > V_{sphere} > V_{cone}. \]