To find the ratio of the volumes of a cone to a cylinder that share the same radius and height, we first need to use the formulas for the volumes of both shapes.
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Volume of a cylinder (V_cylinder): \[ V_{cylinder} = \pi r^2 h \]
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Volume of a cone (V_cone): \[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Given that the height \( h \) is twice the radius \( r \), we substitute \( h = 2r \) into both formulas.
For the cylinder: \[ V_{cylinder} = \pi r^2 (2r) = 2\pi r^3 \]
For the cone: \[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now, we can determine the ratio of the volume of the cone to the volume of the cylinder: \[ \text{Ratio} = \frac{V_{cone}}{V_{cylinder}} = \frac{\frac{2}{3} \pi r^3}{2\pi r^3} \]
The \( \pi r^3 \) terms cancel out: \[ \text{Ratio} = \frac{\frac{2}{3}}{2} = \frac{2}{3} \times \frac{1}{2} = \frac{1}{3} \]
Thus, the ratio of the volume of the cone to the volume of the cylinder is: \[ \frac{1}{3} \]
In summary, the ratio that describes the relationship between the formulas for the volume of a cone to the volume of a cylinder is \( 1:3 \).
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