To approximate the value of \( \sqrt{39} \) using a number line, we need to identify two perfect squares between which 39 lies.
The perfect squares near 39 are:
- \( 6^2 = 36 \)
- \( 7^2 = 49 \)
Thus, \( \sqrt{39} \) is between 6 and 7. To narrow it down further, we can check the squares of numbers between 6 and 7:
- \( 6.3^2 = 39.69 \)
- \( 6.2^2 = 38.44 \)
- \( 6.4^2 = 40.96 \)
From these calculations, we find that:
- \( 6.2^2 < 39 < 6.3^2 \)
- \( 6.3^2 \) is slightly more than 39, and \( 6.2^2 \) is noticeably less.
This suggests that \( \sqrt{39} \) is slightly more than 6.2 but less than 6.3.
To further approximate, we could try:
- \( 6.25^2 = 39.0625 \) (which is slightly greater than 39)
- \( 6.33^2 = 40.0489 \) (which is also greater than 39)
- \( 6.17^2 = 38.4289 \) (which is less than 39)
- \( 6.42^2 = 41.0564 \) (which is greater than 39)
Based on this:
- \( 6.3 \) is too high,
- \( 6.25 \) is the first approximation slightly over 39,
- \( 6.17 \) is lower, and \( 6.33 \) is also too high.
From all our checks, the closest estimates show that \( \sqrt{39} \) is approximately \( 6.24 \) or so.
The selected options are:
- 6.25
- 6.17
- 6.42
- 6.33
Out of these, \( 6.25 \) is the closest approximation to \( \sqrt{39} \). Therefore, the answer is:
6.25