To find the approximate value of [tex]\(\sqrt{19}\)[/tex] using a number line, let's start by understanding the process step-by-step:
1. Identify the exact value of [tex]\(\sqrt{19}\)[/tex]:
We know that [tex]\(\sqrt{19}\)[/tex] is a value between 4 and 5 because:
[tex]\[
4^2 = 16 \quad \text{and} \quad 5^2 = 25
\][/tex]
Since 16 < 19 < 25, it follows that 4 < [tex]\(\sqrt{19}\)[/tex] < 5.
2. Choose the closest values from the provided choices:
The provided choices are:
- 4.5
- 4.25
- 4.13
- 4.38
3. Calculate the differences between each choice and [tex]\(\sqrt{19}\)[/tex]:
The exact value of [tex]\(\sqrt{19}\)[/tex] is approximately 4.3589. We need to compare each choice to this value and find the one that is closest.
- Difference between 4.5 and 4.3589 is:
[tex]\[
|4.5 - 4.3589| = 0.1411
\][/tex]
- Difference between 4.25 and 4.3589 is:
[tex]\[
|4.25 - 4.3589| = 0.1089
\][/tex]
- Difference between 4.13 and 4.3589 is:
[tex]\[
|4.13 - 4.3589| = 0.2289
\][/tex]
- Difference between 4.38 and 4.3589 is:
[tex]\[
|4.38 - 4.3589| = 0.0211
\][/tex]
4. Identify the smallest difference:
We compare these differences:
[tex]\[
0.1411, \quad 0.1089, \quad 0.2289, \quad 0.0211
\][/tex]
The smallest difference is 0.0211, which corresponds to the choice 4.38.
Therefore, using these comparisons, we can conclude that the closest approximate value of [tex]\(\sqrt{19}\)[/tex] from the given choices is:
[tex]\[
\boxed{4.38}
\][/tex]
