To determine the whole number and rational number to the nearest half that \(\sqrt{97}\) is between, we first need to estimate its value.
Calculating \( \sqrt{97} \):
- Find perfect squares near 97:
- \(9^2 = 81\)
- \(10^2 = 100\)
So, we know that: \[ 9 < \sqrt{97} < 10 \]
- Estimate \(\sqrt{97}\):
- To get a more precise estimate, calculate \((9.8)^2\) and \((9.9)^2\):
- \(9.8^2 = 96.04\)
- \(9.9^2 = 98.01\)
- To get a more precise estimate, calculate \((9.8)^2\) and \((9.9)^2\):
This implies: \[ 9.8 < \sqrt{97} < 9.9 \]
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Finding the nearest half:
- The nearest halves around this interval are \(9.5\) and \(10\).
- Since \(9.8\) is closer to \(9.5\) than \(10\), we will choose \(9.5\).
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Identify whole numbers and rational numbers:
- The whole numbers directly below and above \( \sqrt{97} \) are \(9\) and \(10\).
- The rational number to the nearest half is \(9.5\).
Therefore, \(\sqrt{97}\) is between:
- Whole number: 9 and 10
- Rational number to the nearest half: 9.5
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