To find the approximate value of \(\sqrt{19}\) using a number line, we can begin by identifying perfect squares around 19.
The perfect squares nearby are:
- \(4^2 = 16\)
- \(5^2 = 25\)
Since \(19\) is between \(16\) and \(25\), we know that: \[ 4 < \sqrt{19} < 5 \]
Next, to narrow it down further, let's see where \(19\) falls between \(16\) and \(25\).
- The midpoint between \(4\) and \(5\) is \(4.5\).
- Squaring \(4.5\) yields \(20.25\), which is greater than \(19\), meaning \(\sqrt{19}\) must be less than \(4.5\).
- Next, try \(4.4\): \[ 4.4^2 = 19.36 \] This is also more than \(19\), so \(\sqrt{19}\) is less than \(4.4\).
- Then try \(4.3\): \[ 4.3^2 = 18.49 \] This is less than \(19\), meaning \(\sqrt{19}\) is greater than \(4.3\).
Now we have: \[ 4.3 < \sqrt{19} < 4.4 \]
Since the options provided are \(4.13\), \(4.5\), \(4.25\), and \(4.38\), we note that \(4.38\) is the closest approximation in our range.
Thus, the approximate value of \(\sqrt{19}\) is \(4.38\).
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