Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.
Recover cone volume from a matched cylinder
Problem
A cone and matched cylinder \(\text{both}~\text{have}~\text{radius}~4~\text{cm}~\text{and}~\text{height}~9~\text{cm}\), and \(\text{three}~\text{cone}~\text{fills}\) exactly fill the cylinder. Calculate the cylinder volume, apply the \(\text{one}\text{-}\text{third}\) fill ratio, and state the cone's exact volume.
The cone is easiest to measure through its matched cylinder, but the comparison works only when their bases and perpendicular heights agree. We’ll build the shared circular base area, find the cylinder’s volume, translate the stated fill count into a fraction, and apply that fraction to the entire cylinder volume.
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