To determine if Hal made an error in his estimation procedure for \(\sqrt{51.8}\), let's analyze each step:
Step 1: Hal stated that since \(49 < 51.8 < 64\) (where \(7^2 = 49\) and \(8^2 = 64\)), then \(\sqrt{51.8}\) is between 7 and 8. This is correct.
Step 2: Hal indicated that \(51.8\) is closer to \(49\) and suggested squaring the tenths closer to \(7\). However, this sentence is not clearly written. The focus should be on determining the approximate value of \(\sqrt{51.8}\) by estimating rather than squaring. Thus, while he is trying to narrow it down, this step is somewhat unclear. It would be more straightforward to say that he is trying to narrow it down to find the square roots close to \(7.2\) which could be a step in estimating.
Step 3: In this step, he mentions squaring the hundredths closer to \(7.2\). Let's assume he meant to evaluate \(\sqrt{51.8}\) and mention approximating it toward \(7.2\), which is reasonable since \(7.2^2 = 51.84\) and that is close to \(51.8\); this part of reasoning seems coherent.
Step 4: He claims that \(51.8\) is closer to \(51.6961\) than it is to \(51.84\) (where \(7.19^2 = 51.6961\)). Given \(51.8\) is indeed closer to \(51.84\), he incorrectly selects \(7.19\) as the best approximation. The closer approximation should have been \(7.2\) since it squares to \(51.84\) which is actually nearer to \(51.8\).
Conclusion:
Hal made a mistake in Step 4, where he inaccurately selected the approximation by saying \(7.19\) is the best approximation for \(\sqrt{51.8}\). The correct selection should have been \(7.2\), since it offers a closer square to \(51.8\) than \(7.19\).
Correct response: In Step 4, he made an error in selecting the correct decimal approximation.
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