To determine where \( \sqrt{105} \) lies on the number line, we first need to approximate its value.
Calculating \( \sqrt{105} \):
- We know that \( 10^2 = 100 \) and \( 11^2 = 121 \). Therefore, \( \sqrt{105} \) is between 10 and 11.
To find a more precise approximation, we can calculate a few more values:
- \( 10.5^2 = 110.25 \) (which is greater than 105)
- \( 10.2^2 = 104.04 \) (which is less than 105)
- \( 10.3^2 = 106.09 \) (which is greater than 105)
From this, we can conclude that \( \sqrt{105} \) is between 10.2 and 10.3.
We can also see that:
- \( \sqrt{105} \) is approximately 10.24 (considering \( 10.2^2 \) is less than and \( 10.3^2 \) is more than 105).
Now, rounding 10.24 to the nearest half gives us 10.5.
Therefore, \( \sqrt{105} \) lies:
- between 10 and 10.5 when considering the whole number.
- between 10.0 and 10.5 when considering the nearest half rational number.
So the correct response is:
between 10 and 10.5.
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