To find the square root of 105 and determine which interval it falls into, we can first estimate \(\sqrt{105}\).
We know:
- \(\sqrt{100} = 10\)
- \(\sqrt{121} = 11\)
Since \(105\) is between \(100\) and \(121\), we know that \(\sqrt{105}\) will be between \(10\) and \(11\).
To refine our estimate, we can calculate the squares of numbers between \(10\) and \(11\):
- \(10.3^2 = 106.09\) (too high)
- \(10.2^2 = 104.04\) (too low)
- \(10.25^2 = 105.0625\) (too high)
- \(10.24^2 = 104.1876\) (too low)
From our estimates, \(\sqrt{105}\) is slightly more than \(10.2\) but less than \(10.3\).
Since we are looking for the nearest half, we see that \(10.2\) rounds to \(10\) and \(10.3\) rounds to \(10.5\).
Thus, \(\sqrt{105}\) is between \(10\) and \(10.5\).
The correct choice is:
between 10 and 10.5