Question

Using a number line, what is the approximate value of 19−−√ ?(1 point)
Responses

4.13
4.13

4.5
4.5

4.25
4.25

4.38

Answer 1

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 \(4.38\).