To find the approximate value of \( \sqrt{19} \) using a number line, we should first identify two perfect squares that surround the number 19.
The perfect squares around 19 are:
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
Thus, \( \sqrt{19} \) lies between 4 and 5.
To refine our estimate, we can check the squares of numbers between 4 and 5:
- \( 4.3^2 = 18.49 \)
- \( 4.4^2 = 19.36 \)
Since \( 18.49 < 19 < 19.36 \), it follows that \( \sqrt{19} \) lies between 4.3 and 4.4.
For a more precise estimate, let’s refine it further:
- \( 4.35^2 = 18.9225 \)
- \( 4.36^2 = 19.0496 \)
Since \( 18.9225 < 19 < 19.0496 \), we can conclude that \( \sqrt{19} \) is between 4.35 and 4.36.
Thus, the approximate value of \( \sqrt{19} \) is about 4.35.
Among the provided options, the closest value to \( \sqrt{19} \) is 4.38.
So the answer is:
4.38.