To determine the ratio of the volumes of a cone to a cylinder when they share the same radius \( r \) and height \( h \), we first need to use the formulas for the volumes of each shape.
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Volume of a cylinder: \[ V_{\text{cylinder}} = \pi r^2 h \]
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Volume of a cone: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Given that the height \( h \) is twice the radius, we have: \[ h = 2r \]
Now, substituting this expression for height into the volume formulas:
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Volume of the cylinder: \[ V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3 \]
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Volume of the cone: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now, we can find the ratio of the volume of the cone to the volume of the cylinder: \[ \text{Ratio} = \frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{\frac{2}{3} \pi r^3}{2 \pi r^3} \]
The \( \pi r^3 \) terms cancel out: \[ \text{Ratio} = \frac{\frac{2}{3}}{2} = \frac{2}{3} \cdot \frac{1}{2} = \frac{1}{3} \]
Thus, the ratio that describes the relationship between the volumes of the cone and the cylinder is: \[ \frac{1}{3} \]
In conclusion, the ratio of the volume of the cone to the volume of the cylinder is \( \frac{1}{3} \).