To determine the whole number and rational number to the nearest half that \(\sqrt{105}\) lies between, we first need to find the approximate value of \(\sqrt{105}\).
Calculating \(\sqrt{105}\):
We know that:
- \(\sqrt{100} = 10\)
- \(\sqrt{121} = 11\)
Since \(105\) is between \(100\) and \(121\), it follows that: \[ 10 < \sqrt{105} < 11 \]
Next, we can estimate \(\sqrt{105}\) more precisely. Testing the midpoint between 10 and 11:
- \(10.5^2 = 110.25\) (too high)
- \(10.25^2 = 105.0625\) (too high)
We can find that:
- \(10.2^2 = 104.04\) (too low)
- \(10.3^2 = 106.09\) (too high)
This gives us a better estimate: \[ 10.2 < \sqrt{105} < 10.3 \]
When rounding to the nearest half, we consider the intervals:
- \(10\)
- \(10.5\)
- \(11\)
Thus, \(\sqrt{105}\) is greater than \(10.2\) and less than \(10.3\), so it rounds to \(10.5\).
In conclusion:
- The whole number that \(\sqrt{105}\) is between is \(10\) and \(11\).
- The rational number to the nearest half that \(\sqrt{105}\) is between is \(10.5\).
Therefore, the correct response is: between 10 and 10.5.
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