To approximate the value of \(\sqrt{19}\) using a number line, we can first identify the squares of whole numbers around 19.
- \(4^2 = 16\)
- \(5^2 = 25\)
Since \(19\) is between \(16\) and \(25\), we know that:
\[ 4 < \sqrt{19} < 5 \]
Next, we can check some decimal values between \(4\) and \(5\):
- \(4.25^2 = 18.0625\)
- \(4.5^2 = 20.25\)
- \(4.4^2 = 19.36\)
- \(4.38^2 \approx 19.0244\)
- \(4.3^2 = 18.49\)
From these calculations, we can see that:
- \(4.38^2\) is a close approximation, being just above \(19\).
- \(4.4^2\) is above \(19\), while \(4.3^2\) is below \(19\).
Given these estimates, the best choice among the options provided is \(C. 4.38\), as it gives a value very close to \(\sqrt{19}\).
Thus, the approximate value of \(\sqrt{19}\) is:
C. 4.38