To answer the question of which whole number and rational number to the nearest half [tex]$\sqrt{97}$[/tex] is between, let's follow these steps:
1. Calculate the square root of 97:
- The square root of 97 is approximately [tex]\(9.848857801796104\)[/tex].
2. Identify the whole numbers it lies between:
- The floor value (the greatest integer less than or equal to [tex]$\sqrt{97}$[/tex]) is [tex]\(9\)[/tex].
- The ceiling value (the smallest integer greater than or equal to [tex]$\sqrt{97}$[/tex]) is [tex]\(10\)[/tex].
- Therefore, [tex]$\sqrt{97}$[/tex] is between the whole numbers [tex]\(9\)[/tex] and [tex]\(10\)[/tex].
3. Identify the nearest half number:
- Consider the nearest half numbers between [tex]\(9\)[/tex] and [tex]\(10\)[/tex]. These would be [tex]\(9.5\)[/tex] (halfway between [tex]\(9\)[/tex] and [tex]\(10\)[/tex]).
- Compare [tex]\((\sqrt{97} - 9)\)[/tex] and [tex]\((10 - \sqrt{97})\)[/tex].
- [tex]\(\sqrt{97} - 9 \approx 0.848857801796104\)[/tex]
- [tex]\(10 - \sqrt{97} \approx 0.151142198203896\)[/tex]
- Since [tex]\(0.151142198203896 < 0.5\)[/tex], [tex]\(\sqrt{97}\)[/tex] is closer to [tex]\(9.5\)[/tex] than [tex]\(10\)[/tex].
By considering the positions on a number line and the closest half point:
- [tex]\(\sqrt{97}\)[/tex] is between the whole numbers 9 and 10.
- The nearest rational number to the nearest half would be 9.5.
Therefore,
[tex]\[
\sqrt{97} \text{ is between } 9 \text{ and } 10.
\][/tex]
To the nearest half:
[tex]\[
\sqrt{97} \text{ is closest to } 9.5.
\][/tex]
