To determine the approximate value of [tex]\(\sqrt{19}\)[/tex] using a number line and provided options, let's examine the approximate value of [tex]\(\sqrt{19}\)[/tex]. We need to place [tex]\(\sqrt{19}\)[/tex] between two integers and compare it with the given options.
Firstly, we recall that:
[tex]\[
\sqrt{16} = 4
\][/tex]
[tex]\[
\sqrt{25} = 5
\][/tex]
Since [tex]\(19\)[/tex] is between [tex]\(16\)[/tex] and [tex]\(25\)[/tex], it follows that:
[tex]\[
4 < \sqrt{19} < 5
\][/tex]
To narrow it down further, let's examine the given multiple-choice options:
- 14.13
- 4.5
- 4.38
- 4.25
Among these options, only [tex]\(4.38\)[/tex] is plausible as it's located between 4 and 5.
By testing or comparing these values, we find that the value of [tex]\(\sqrt{19}\)[/tex] is best approximated by:
[tex]\[
\sqrt{19} \approx 4.358898943540674
\][/tex]
Thus, the value closest to this approximation from the given choices is [tex]\(4.38\)[/tex].
Therefore, the approximate value of [tex]\(\sqrt{19}\)[/tex] is [tex]\(4.38\)[/tex].
