To determine between which two numbers the square root of 59 lies, we can first find the perfect squares around 59. The perfect squares near 59 are:
- \(7^2 = 49\)
- \(8^2 = 64\)
Since \(49 < 59 < 64\), we can conclude that:
\[ 7 < \sqrt{59} < 8 \]
Next, we can narrow it down further by checking the midpoints:
- \(7.5^2 = 56.25\)
- \(8^2 = 64\)
Since \(56.25 < 59 < 64\), this implies that:
\[ 7.5 < \sqrt{59} < 8 \]
Now, we can compare this to the provided options for rational numbers rounded to the nearest half:
- 8.5 and 9
- 6 and 6.5
- 7 and 7.5
- 7.5 and 8
From our evaluation, we found that \(\sqrt{59}\) is between 7.5 and 8. Therefore, the correct choice is:
7.5 and 8.