To determine where \( \sqrt{58} \) lies on a number line, we first need to calculate \( \sqrt{58} \).
To approximate \( \sqrt{58} \):
- We know that \( 7^2 = 49 \) and \( 8^2 = 64 \).
- Thus, \( 7 < \sqrt{58} < 8 \).
For a more accurate approximation, we can calculate:
- \( 7.5^2 = 56.25 \)
- \( 7.6^2 = 57.76 \)
- \( 7.7^2 = 59.29 \)
Since \( 56.25 < 58 < 59.29 \), we know that \( 7.5 < \sqrt{58} < 7.7 \).
To estimate \( \sqrt{58} \) to the nearest half, we can see that:
- Since \( 7.5 < \sqrt{58} < 7.6 \), we round \( \sqrt{58} \) to approximately \( 7.6 \).
Checking for the nearest half, \( 7.5 \) and \( 8 \) would be the rational numbers around it.
Thus, \( \sqrt{58} \) is between \( 7.5 \) and \( 8 \).
The correct response is: between 7.5 and 8.