To determine the interval in which the square root of [tex]\(105\)[/tex] falls, we'll first approximate the value of [tex]\(\sqrt{105}\)[/tex].
The square root of [tex]\( 105 \)[/tex] is:
[tex]\[ \sqrt{105} \approx 10.246950765959598 \][/tex]
Next, let's identify the interval within which this value falls on a number line. We need to compare it to commonly known maximum and minimum values forming halves or whole values.
1. The interval between 9.5 and 10 can be tested:
- [tex]\(9.5 \leq 10.246950765959598 < 10\)[/tex]
- [tex]\(10.246950765959598\)[/tex] is greater than 10, so this interval is incorrect.
2. The interval between 10 and 10.5 can be tested:
- [tex]\(10 \leq 10.246950765959598 < 10.5\)[/tex]
- [tex]\(10.246950765959598\)[/tex] is within this interval as it is greater than 10 and less than 10.5. Thus, this interval is correct.
3. The interval between 10.5 and 11 can be tested:
- [tex]\(10.5 \leq 10.246950765959598 < 11\)[/tex]
- [tex]\(10.246950765959598\)[/tex] is less than 10.5, so this interval is incorrect.
4. The interval between 11 and 11.5 can be tested:
- [tex]\(11 \leq 10.246950765959598 < 11.5\)[/tex]
- [tex]\(10.246950765959598\)[/tex] is less than 11, so this interval is incorrect.
Based on the comparisons above, the square root of [tex]\(105\)[/tex] falls between [tex]\(10\)[/tex] and [tex]\(10.5\)[/tex].
Thus, the correct answer is:
- between 10 and 10.5.
