Problem

Using \(\text{Cavalieri}'s~\text{principle}\), what can you conclude about the volumes of \(\text{two}~\text{prisms}\) with equal height and equal cross-sectional area at every level?

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What this problem is really about

Cavalieri’s principle compares solids layer by layer, so matching overall shapes is not the relevant test. We’ll verify the common height, compare cross-sectional areas at every corresponding level, imagine equal-thickness layers, and use their accumulated contributions to determine the volume comparison.

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By Cavalieri’s principle, the two prisms cannot be compared unless their base areas are equal, not just their cross-sections at every level.

By Cavalieri’s principle, the two prisms must have the same volume because they have equal height and equal cross-sectional area at every level.

By Cavalieri’s principle, the two prisms must have the same volume only if they also have the same overall shape.

By Cavalieri’s principle, the two prisms may still have different volumes because equal slices at each level do not determine the whole solid.

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Using \(\text{Cavalieri}'s~\text{principle}\), what can you conclude about the volumes of \(\text{two}~\text{prisms}\) with equal height and equal cross-sectional area at every level?

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