To determine between which whole number and rational number \( \sqrt{105} \) lies, we first need to calculate the value of \( \sqrt{105} \).
Calculating \( \sqrt{105} \):
- Estimate the square roots of nearby whole numbers:
- \( 10^2 = 100 \)
- \( 11^2 = 121 \)
This means \( 10 < \sqrt{105} < 11 \).
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Finding a more accurate estimate:
- \( 10.5^2 = 110.25 \), so \( \sqrt{105} < 10.5 \).
- \( 10.2^2 = 104.04 \) and \( 10.3^2 = 106.09 \), indicating \( 10.2 < \sqrt{105} < 10.3 \).
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Conclusion about location:
- Since \( \sqrt{105} \) is about 10.24 (when calculated more precisely), we find this value lies between \( 10.2 \) (which rounds to 10.5) and \( 10.3 \).
From the analysis above, we conclude that:
\( \sqrt{105} \) is between 10 and 10.5 when rounded to the nearest half.
Correct response: between 10 and 10.5