To approximate \( \sqrt{1131} \) and get closer to \( \sqrt{31} \) (since \( 31^2 = 961 \) and \( 32^2 = 1024 \), meaning \( \sqrt{1031} \) is around 31.3 and is actually closer to \( 33 \)), we can use a sequence of approximations.
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Initial Guess: Start by estimating \( \sqrt{1131} \). Notice that:
- \( 30^2 = 900 \)
- \( 31^2 = 961 \)
- \( 32^2 = 1024 \)
- \( 33^2 = 1089 \)
- \( 34^2 = 1156 \)
This gives us a rough estimate that \( \sqrt{1131} \) is between \( 33 \) and \( 34 \).
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First Approximation: Let's start with \( 33.5 \).
\[ 33.5^2 = 1122.25 \]
This is less than \( 1131 \), so we increase our guess.
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Second Approximation: Try \( 33.6 \).
\[ 33.6^2 = 1129.76 \]
Still less than \( 1131 \), so we need a higher value.
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Third Approximation: Now, try \( 33.7 \).
\[ 33.7^2 = 1135.69 \]
This is more than \( 1131 \), indicating our approximation is between \( 33.6 \) and \( 33.7 \).
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Fourth Approximation: We can try the average of \( 33.6 \) and \( 33.7 \), which is \( 33.65 \).
\[ 33.65^2 = 1132.5225 \]
Again, this is more than \( 1131 \), so we can narrow it down.
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Fifth Approximation: Try \( 33.64 \).
\[ 33.64^2 \approx 1131.7696 \]
This is again a bit more than \( 1131 \).
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Sixth Approximation: Now let’s try \( 33.63 \).
\[ 33.63^2 \approx 1130.5969 \]
This is less than \( 1131 \).
Now we know that:
- \( 33.63^2 < 1131 < 33.64^2 \)
To approximate to one decimal place, we can summarize our findings:
Since \( 33.63 < \sqrt{1131} < 33.64 \), we conclude that:
\[ \sqrt{1131} \approx 33.6 \text{ (to one decimal place)} \]
Therefore, the answer is:
\[ \sqrt{1131} \approx 33.6 \]
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