To find the whole number and the rational number to the nearest half that \( \sqrt{97} \) is between, we first need to calculate the approximate value of \( \sqrt{97} \).
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Estimate \( \sqrt{97} \):
- We know \( 9^2 = 81 \) and \( 10^2 = 100 \), so \( 9 < \sqrt{97} < 10 \).
- A more precise calculation can be done using a calculator or by approximation methods. We find: \[ \sqrt{97} \approx 9.8488 \]
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Identify the whole number:
- The whole number immediately less than \( 9.8488 \) is \( 9 \) and the one immediately greater is \( 10 \).
- Therefore, \( \sqrt{97} \) is between the whole numbers \( 9 \) and \( 10 \).
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Rational number to the nearest half:
- The nearest half values around \( 9.8488 \) are \( 9.5 \) and \( 10.0 \).
- Since \( 9.8488 \) is closer to \( 10.0 \) than it is to \( 9.5 \), the rational number to the nearest half is \( 10.0 \).
Conclusion: The whole number \( \sqrt{97} \) is between is \( 9 \) and \( 10 \), and the rational number to the nearest half is \( 10.0 \).