To approximate the value of \( \sqrt{19} \) on a number line, we can find two perfect squares between which 19 falls.
The perfect squares near 19 are:
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
This indicates that \( 4 < \sqrt{19} < 5 \).
Next, we can check the squares of numbers between 4 and 5 to narrow it down further:
- \( 4.3^2 = 18.49 \)
- \( 4.4^2 = 19.36 \)
Since \( 18.49 < 19 < 19.36 \), we know: \( 4.3 < \sqrt{19} < 4.4 \).
Now let's check 4.35:
- \( 4.35^2 = 18.9225 \)
Since \( 18.9225 < 19 \), we can conclude: \( 4.35 < \sqrt{19} < 4.4 \).
Now let's check 4.37:
- \( 4.37^2 = 19.0569 \)
Since \( 19.0569 > 19 \), we have: \( 4.35 < \sqrt{19} < 4.37 \).
Now let’s check 4.38:
- \( 4.38^2 = 19.1844 \)
Since \( 19.1844 > 19 \), we establish: \( 4.35 < \sqrt{19} < 4.38 \).
Therefore, the best approximation can be selected from the options given:
- A. 4.38
- B. 4.25
Since \( \sqrt{19} \) is approximately between 4.35 and 4.38, the better option is:
A. 4.38.