To determine the relationship between the volumes of the sphere, cylinder, and cone with the specified characteristics, we first need the volume formulas for each shape:
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Sphere: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
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Cylinder: \[ V_{\text{cylinder}} = \pi r^2 h \] Given that the height \(h\) is twice the radius \(r\), we substitute \(h = 2r\): \[ V_{\text{cylinder}} = \pi r^2 (2r) = 2 \pi r^3 \]
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Cone: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Again substituting \(h = 2r\): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we can compare the volumes:
- \( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \)
- \( V_{\text{cylinder}} = 2 \pi r^3 \)
- \( V_{\text{cone}} = \frac{2}{3} \pi r^3 \)
To analyze the relationships, we can express the volumes in terms of fractions of \( V_{\text{cylinder}} \):
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The volume of the sphere compared to the cylinder: \[ \frac{V_{\text{sphere}}}{V_{\text{cylinder}}} = \frac{\frac{4}{3} \pi r^3}{2 \pi r^3} = \frac{4/3}{2} = \frac{4}{6} = \frac{2}{3} \] So: \[ V_{\text{sphere}} = \frac{2}{3} V_{\text{cylinder}} \]
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The volume of the cone compared to the cylinder: \[ \frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{\frac{2}{3} \pi r^3}{2 \pi r^3} = \frac{2/3}{2} = \frac{2}{6} = \frac{1}{3} \] So: \[ V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} \]
In summary, the relationships are:
- The volume of the sphere is \(\frac{2}{3}\) of the volume of the cylinder.
- The volume of the cone is \(\frac{1}{3}\) of the volume of the cylinder.
Thus, the relationships can be expressed as: \[ V_{\text{sphere}} < V_{\text{cone}} < V_{\text{cylinder}} \] Specifically, in fractional form: \[ V_{\text{sphere}} = \frac{2}{3} V_{\text{cylinder}} \quad \text{and} \quad V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} \] So the volume of the sphere is greater than the volume of the cone and less than that of the cylinder.