To find the ratio of the volume of a cone to the volume of a sphere when both have the same radius \( r \), and the height of the cone is twice the radius \( (h = 2r) \), we first need to calculate the volumes of both shapes using their respective formulas.
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Volume of a Sphere: The formula for the volume of a sphere is given by: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
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Volume of a Cone: The formula for the volume of a cone is given by: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Since the height \( h \) of the cone is twice the radius \( r \), we substitute \( h \) with \( 2r \): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now that we have both volumes, we can find the ratio of the volume of the cone to the volume of the sphere: \[ \text{Ratio} = \frac{V_{\text{cone}}}{V_{\text{sphere}}} = \frac{\frac{2}{3} \pi r^3}{\frac{4}{3} \pi r^3} \]
First, let's simplify this ratio: \[ = \frac{2/3}{4/3} = \frac{2}{4} = \frac{1}{2} \]
Therefore, the ratio of the volume of the cone to the volume of the sphere is \( \frac{1}{2} \).
Explanation:
This ratio describes the relationship between the two volumes: the volume of the cone is half that of the sphere when both share the same radius and the cone's height is twice that radius. This is due to the geometric properties of these shapes and their volume formulas. The linear dimensions (height and radius) directly affect their volume, and in this specific case, the increase in height of the cone beyond what is typical for a cone (which typically has a height less than or equal to its diameter) influences the final ratio by keeping the volume of the cone considerably smaller than that of the sphere.