To derive the volume of a cone based on the volume of a cylinder, you can think of how the shapes relate to each other. A cone fits nicely into a cylinder that has the same base radius and height.
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Understanding the Shapes: Imagine a cylinder and a cone with the same height \( h \) and the same base radius \( r \). The cylinder completely encloses the cone.
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Visualizing the Volume: If you fill the cylinder with a liquid, the cone occupies part of that space. Through experimentation (like using physical models), one can discover that the volume of the cone is actually only a fraction of the volume of the cylinder.
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Finding the Relationship: It’s been established (through geometry) that the volume of the cone is exactly one-third of the volume of the cylinder when both share the same height and base. If the volume of the cylinder is given by \( V_{cylinder} = \pi r^2 h \), the volume of the cone can be derived as \( V_{cone} = \frac{1}{3} \pi r^2 h \).
Therefore, to find the volume of the cone, you need to multiply the volume of the cylinder by \( \frac{1}{3} \).
The correct response is: You need to multiply the volume of the cylinder by \( \frac{1}{3} \).