To find the ratio of the volume of a cone to the volume of a cylinder, we need to start with the formulas for their volumes.
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Volume of a Cylinder (V_cylinder): \[ V_{\text{cylinder}} = \pi r^2 h \] Here, \(r\) is the radius and \(h\) is the height.
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Volume of a Cone (V_cone): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Given that the height of both the cylinder and the cone is twice the radius (so \(h = 2r\)), we can substitute this into the formulas.
Without substitution:
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For the cylinder: \[ V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3 \]
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For the cone: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now, we can determine the ratio of the volumes: \[ \text{Ratio} = \frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{\frac{2}{3} \pi r^3}{2 \pi r^3} \] Simplifying this gives: \[ \text{Ratio} = \frac{\frac{2}{3}}{2} = \frac{2}{3} \cdot \frac{1}{2} = \frac{1}{3} \]
Thus, the ratio that describes the relationship between the volume of the cone and the volume of the cylinder is: \[ \frac{1}{3} \]