To find the approximate value of \(\sqrt{19}\), let's analyze the values of perfect squares around 19:
- \(4^2 = 16\)
- \(5^2 = 25\)
This means \(\sqrt{19}\) is between 4 and 5.
Next, to narrow it down further:
- \(4.3^2 = 18.49\)
- \(4.4^2 = 19.36\)
Thus, \(\sqrt{19}\) is between 4.3 and 4.4.
Now, checking a value in the middle:
- \(4.35^2 = 18.9225\) (is less than 19)
- \(4.36^2 = 19.0496\) (is greater than 19)
Hence, \(\sqrt{19}\) is approximately 4.36.
Among the options given, 4.38 is the closest approximation.
The answer is 4.38.