To find the whole number and the rational number to the nearest half that the square root of 105 falls between, let's first estimate \(\sqrt{105}\).
We know that:
- \(\sqrt{100} = 10\)
- \(\sqrt{121} = 11\)
Since \(105\) is between \(100\) and \(121\), it follows that: \[ 10 < \sqrt{105} < 11 \]
Next, let's approximate \(\sqrt{105}\) more closely. Squaring some fractions can help us narrow down the value:
- \(10.2^2 = 104.04\)
- \(10.3^2 = 106.09\)
From this, we see that: \[ 10.2 < \sqrt{105} < 10.3 \]
To find the nearest half, we can consider the halves around \(10\):
- \(10\)
- \(10.5\)
Since \(10.2\) is closer to \(10.5\) than to \(10\), we round \(\sqrt{105}\) to: \[ 10.5 \]
Thus, the square root of \(105\) is between the whole number \(10\) and the rational number \(10.5\) when rounded to the nearest half.