To find the ratio between the volumes of a cone and a cylinder that share the same radius \( r \) and height \( h \) (where \( h = 2r \)), we can start by using the formulas for each shape's volume.
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Volume of the Cylinder: \[ V_{cylinder} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]
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Volume of the Cone: \[ V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now, to find the ratio of the volume of the cone to the volume of the cylinder: \[ \text{Ratio} = \frac{V_{cone}}{V_{cylinder}} = \frac{\frac{2}{3} \pi r^3}{2 \pi r^3} \]
This simplifies as follows: \[ \text{Ratio} = \frac{\frac{2}{3}}{2} = \frac{2}{3} \cdot \frac{1}{2} = \frac{1}{3} \]
Therefore, the ratio of the volume of the cone to the volume of the cylinder is: \[ \frac{1}{3} : 1 \]
So the final answer for the ratio in the format (Blank):(Blank) is: 1:3