To find the approximate value of \( \sqrt{39} \) on a number line, we can consider the squares of whole numbers close to 39.
- \(6^2 = 36\)
- \(7^2 = 49\)
Since \( 39 \) is between \( 36 \) and \( 49 \), we know that \( \sqrt{39} \) is between \( 6 \) and \( 7 \). Next, we can narrow it down further:
- Calculate \( 6.2^2 = 38.44 \)
- Calculate \( 6.3^2 = 39.69 \)
Now we see that:
- \( 6.2^2 < 39 < 6.3^2 \)
This means \( \sqrt{39} \) is between \( 6.2 \) and \( 6.3 \). Since \( 39 \) is closer to \( 36 \) than to \( 49 \), we can estimate that \( \sqrt{39} \) is closer to \( 6.2 \).
To check values closer to \( 6.3 \):
- Calculate \( 6.25^2 = 39.0625 \)
Now we can see that \( 6.25^2 \) is slightly greater than \( 39 \):
- Since \( 39 < 6.25^2 \) and \( 6.2^2 < 39 \), \( \sqrt{39} \) must be somewhere between \( 6.2 \) and \( 6.25 \).
Thus, the best approximation from the given options is 6.25.