To find the values that \( \sqrt{58} \) lies between, we first need to approximate \( \sqrt{58} \).
Calculating:
- Since \( 7^2 = 49 \) and \( 8^2 = 64 \), it follows that \( 7 < \sqrt{58} < 8 \).
To narrow it down further, we can estimate:
- \( 7.5^2 = 56.25 \) (which is less than 58)
- \( 7.6^2 = 57.76 \) (which is also less than 58)
- \( 7.7^2 = 59.29 \) (which is greater than 58)
From this, we can conclude:
\[ 7.6 < \sqrt{58} < 7.7 \]
Next, we find the rational numbers to the nearest half. The closest halves to consider are:
- 7.5
- 8.0
Therefore, \( \sqrt{58} \) is between \( 7.5 \) and \( 7.7 \), which means:
- The whole number \( \sqrt{58} \) lies between is 7.
- The rational number to the nearest half is 7.5.
In summary, \( \sqrt{58} \) is between the whole number 7 and the rational number 7.5.