Problem

A filled right triangle with legs \(a\) and \(b\) is rotated separately about leg \(a\) and about leg \(b\). Use the diagrams to identify \(\text{both}~\text{solids}\), their radii and heights, their slant heights, and whether their volumes must match.

Prompt diagram for M3-037-A07-V02: the generating region, rotation axis, dimensions, and 360-degree sweep with the resulting solid withheld.

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The results are two cones: about a, radius b and height a; about b, radius a and height b. Both have slant height sqrt(a²+b²), and their volumes are always equal.

The results are two cones: about a, radius b and height a; about b, radius a and height b. Both have slant height sqrt(a²+b²), and their volumes need not match.

The results are two cylinders: about a, radius b and height a; about b, radius a and height b. Their volumes are pi b² a and pi a² b, respectively.

The results are two cones: about a, radius a and height b; about b, radius b and height a. Both have slant height sqrt(a²+b²), and their volumes need not match.

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A filled right triangle with legs \(a\) and \(b\) is rotated separately about leg \(a\) and about leg \(b\). Use the diagrams to identify \(\text{both}~\text{solids}\), their radii and heights, their slant heights, and whether their volumes must match.

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