Problem

A rectangle has side lengths \(x\) and \(x+5\) and area \(84\). Model A is \(x+(x+5)=84\), and Model B is \(x(x+5)=84\). Evaluate \(\text{both}~\text{models}\) against the context. Report the valid model, the governing relationship, the feature that matches, and the mismatch in the rejected model.

Big Picture

What this problem is really about

Before comparing the proposed models, we need the governing relationship from the context. We’ll use the fact that rectangle area multiplies two side lengths, then check each model’s operation and units; the model whose structure gives square units represents area, while adding the sides represents a different geometric quantity.

Answer choices

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selected model=A; governing relationship=area=length+width; feature match=add x and x+5; rejected-model mismatch=B multiplies the side lengths instead of combining them by addition

selected model=A; governing relationship=perimeter=length+width; feature match=add x and x+5; rejected-model mismatch=B finds area rather than perimeter

selected model=B; governing relationship=area=length·width; feature match=multiply x and x+5; rejected-model mismatch=A adds side lengths and would model half-perimeter, not area

selected model=B; governing relationship=area=length·width; feature match=x(x+5)=x²+5; rejected-model mismatch=A adds the side lengths instead of multiplying them

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A rectangle has side lengths \(x\) and \(x+5\) and area \(84\). Model A is \(x+(x+5)=84\), and Model B is \(x(x+5)=84\). Evaluate \(\text{both}~\text{models}\) against the context. Report the valid model, the governing relationship, the feature that matches, and the mismatch in the rejected model.

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