To find the ratio between the volume of a cone and the volume of a sphere when they share the same radius \( r \) and the height of the cone is twice the radius, we can use the formulas for the volumes of both shapes.
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Volume of the Cone \( V_c \): The formula for the volume of a cone is given by: \[ V_c = \frac{1}{3} \pi r^2 h \] Since the height \( h \) of the cone is twice the radius, we have: \[ h = 2r \] Substituting \( h \) into the cone volume formula: \[ V_c = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
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Volume of the Sphere \( V_s \): The formula for the volume of a sphere is given by: \[ V_s = \frac{4}{3} \pi r^3 \]
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Finding the Ratio: Now we want to find the ratio of the volume of the cone to the volume of the sphere: \[ \text{Ratio} = \frac{V_c}{V_s} = \frac{\frac{2}{3} \pi r^3}{\frac{4}{3} \pi r^3} \] The \( \pi r^3 \) terms cancel out: \[ \text{Ratio} = \frac{\frac{2}{3}}{\frac{4}{3}} = \frac{2}{4} = \frac{1}{2} \]
Thus, the ratio describing the relationship between the volumes of the cone to the sphere is \( \frac{1}{2} \).
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