To determine the whole number and rational number to the nearest half that [tex]\(\sqrt{105}\)[/tex] is between, we first need to find an approximation of [tex]\(\sqrt{105}\)[/tex].
Given that [tex]\(\sqrt{105} \approx 10.246950765959598\)[/tex], we can place this value on a number line.
Here are the major steps to identify the correct interval:
1. Whole Numbers: Look for the nearest whole numbers between which [tex]\(\sqrt{105}\)[/tex] lies.
- The nearest whole numbers are 10 and 11, since [tex]\(10 < 10.246950765959598 < 11\)[/tex].
2. Rational Numbers to the Nearest Half: Identify the rational numbers to the nearest half that bound [tex]\(\sqrt{105}\)[/tex].
- Evaluating the halves, we have 10.0, 10.5, 11.0, etc.
- Since [tex]\(10 < 10.246950765959598 < 10.5\)[/tex] and [tex]\(10.246950765959598 < 11\)[/tex], we compare the value closer to the halves.
- Here, [tex]\(10.246950765959598\)[/tex] is greater than 10 but less than 10.5 and less than 11.
Given that [tex]\(\sqrt{105}\)[/tex] lies between 10.5 and 11:
The answer is:
Between 10.5 and 11.
