To find the ratio of the volumes of the cone and the sphere given that they share the same radius (r) and the height of the cone is twice the radius (h = 2r), we can use the formulas for the volumes of a cone and a sphere.
The volume \( V \) of a cone is given by the formula: \[ V_{cone} = \frac{1}{3} \pi r^2 h \] Since \( h = 2r \), we can substitute that into the formula: \[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
The volume \( V \) of a sphere is given by the formula: \[ V_{sphere} = \frac{4}{3} \pi r^3 \]
Now, we can form the ratio of the volume of the cone to the volume of the sphere: \[ \text{Ratio} = \frac{V_{cone}}{V_{sphere}} = \frac{\frac{2}{3} \pi r^3}{\frac{4}{3} \pi r^3} \]
The \( \frac{4}{3} \pi r^3 \) in the denominator will cancel out with the \( \frac{4}{3} \pi r^3 \) in the numerator, leading to: \[ \text{Ratio} = \frac{2}{4} = \frac{1}{2} \]
Thus, the ratio of the volumes of the cone to the sphere is: \[ \frac{1}{2} \]
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