Using a number line, what is the approximate value of 19−−√?

A. 4.38
B . 4.25

1 answer

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.