Problem

Situation A paints only the triangular faces of a square pyramid with base side \(6~m\) and slant height \(5~m\). Situation B wraps the entire outside of a \(6-8-10\) triangular prism of length \(4~m\). Which complete response correctly compares the surface-area quantities?

A square pyramid with only four triangular faces included beside a triangular prism with its entire closed outside included.

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Situation A: area of the painted faces = \(\tfrac12Pl = \tfrac12(24)(5)=60\text{ m}^2\). Situation B: surface area of the prism = \(Ph + 2B = (24)(4)+2(24)=144\text{ m}^2\). Judgment: same measurement type because both answers are in square meters.

Situation A: total surface area; SA = \(\tfrac12Pl + B\); \(P = 4(6)=24\) m, so \(SA = \tfrac12(24)(5) + 6^2 = 96\text{ m}^2\). Situation B: total surface area; \(SA = Ph + 2B\); \(B = \tfrac12(6)(8)=24\text{ m}^2\) and \(P=6+8+10=24\) m, so \(SA=(24)(4)+2(24)=144\text{ m}^2\). Judgment: same measurement type.

Situation A: lateral surface area; \(LSA = \tfrac12Pl\); \(P = 4(6)=24\) m, so \(LSA = \tfrac12(24)(5)=60\text{ m}^2\). Situation B: total surface area; \(SA = Ph + 2B\); \(B = \tfrac12(6)(8)=24\text{ m}^2\) and \(P=6+8+10=24\) m, so \(SA=(24)(4)+2(24)=144\text{ m}^2\). Judgment: different measurement types.

Situation A: lateral surface area; \(LSA = \tfrac12Pl = \tfrac12(24)(5)=60\text{ m}^2\). Situation B: lateral surface area; \(LA = Ph = (24)(4)=96\text{ m}^2\). Judgment: same measurement type.

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Situation A paints only the triangular faces of a square pyramid with base side \(6~m\) and slant height \(5~m\). Situation B wraps the entire outside of a \(6-8-10\) triangular prism of length \(4~m\). Which complete response correctly compares the surface-area quantities?

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