To find the whole number and rational number to the nearest half that \( \sqrt{97} \) is between, we first need to estimate \( \sqrt{97} \).
Calculating some perfect squares around 97:
- \( 9^2 = 81 \)
- \( 10^2 = 100 \)
Since \( 97 \) is between \( 81 \) and \( 100 \), \( \sqrt{97} \) is between \( 9 \) and \( 10 \).
To get a more precise estimate, we can calculate:
\[ 9.5^2 = 90.25 \] \[ 9.6^2 = 92.16 \] \[ 9.7^2 = 94.09 \] \[ 9.8^2 = 96.04 \] \[ 9.9^2 = 98.01 \]
From this, we see that \( \sqrt{97} \) is between \( 9.8 \) and \( 9.9 \).
To round \( \sqrt{97} \) to the nearest half, it is simply between \( 9.5 \) and \( 10 \). Therefore, rounding to the nearest half gives us \( 9.5 \) as a rational number.
In conclusion, \[ \sqrt{97} \text{ is between } 9 \text{ and } 9.5. \]