To compare the volumes of a cone and a cylinder that share the same radius \( r \) and height \( h \), we start with their respective volume formulas.
Volume of the Cylinder: The formula for the volume \( V_c \) of a cylinder is given by: \[ V_c = \pi r^2 h \]
Volume of the Cone: The formula for the volume \( V_{cone} \) of a cone is: \[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Now, in this case, we know that the height \( h \) is twice the radius, so we can express \( h \) as: \[ h = 2r \]
Now, substituting \( h \) into the formulas of the volumes:
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Volume of the Cylinder: \[ V_c = \pi r^2 (2r) = 2\pi r^3 \]
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Volume of the Cone: \[ V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we can find the ratio of the volume of the cone to the volume of the cylinder: \[ \text{Ratio} = \frac{V_{cone}}{V_c} = \frac{\frac{2}{3} \pi r^3}{2\pi r^3} \]
Cancelling out \( \pi r^3 \) from both the numerator and the denominator, we get: \[ \text{Ratio} = \frac{\frac{2}{3}}{2} = \frac{2}{3} \cdot \frac{1}{2} = \frac{1}{3} \]
Thus, the ratio of the volume of the cone to the volume of the cylinder is \( \frac{1}{3} \).
Explanation:
This ratio indicates that the volume of a cone is one-third the volume of a cylinder when they share the same radius and height. This relationship stems from the geometry of these shapes, specifically how the cone's volume is derived as a third of the pyramid-like structure compared to the full space occupied by the cylinder. The cone, having a tapered shape, contains significantly less volume, leading to this consistent ratio of \( \frac{1}{3} \).
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