Hal did not make an error. However, let's clarify the details:
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Step 1: Hal correctly stated that since \( 9^2 = 81 \) and \( 10^2 = 100 \), \( \sqrt{82.5} \) is indeed between 9 and 10.
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Step 2: Hal calculated the squares of \( 9.0 \), \( 9.1 \), and \( 9.2 \) correctly. He noted that \( 81 < 82.5 < 82.81 \), which is accurate.
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Step 3: Hal proceeded to square values closer to \( 9.1 \), \( 9.08 \) and \( 9.09 \), which correctly bracket \( 82.5 \) with \( 82.44 < 82.5 < 82.62 \).
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Step 4: Since \( 82.5 \) is closer to \( 82.44 \) than to \( 82.62 \), Hal's estimation of \( 9.09 \) as the best approximation is indeed valid.
Given this analysis, Hal did not make any errors in his estimation process. Therefore, the correct response is:
Hal did not make an error.