To find the whole number and rational number to the nearest half that the expression \(\sqrt{40}\) is between, we first need to evaluate the value of \(\sqrt{40}\).
Calculating \(\sqrt{40}\):
\[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10} \]
Next, we approximate \(\sqrt{10}\):
- Since \(3^2 = 9\) and \(4^2 = 16\), we know that \(\sqrt{10}\) is between 3 and 4.
- A more precise approximation of \(\sqrt{10}\) is about 3.162. Therefore:
\[ \sqrt{40} \approx 2 \times 3.162 = 6.324 \]
Now, identifying the whole number and rational number to the nearest half:
- Whole number: The value \(6.324\) is between 6 and 7.
- Nearest half: Rounding \(6.324\) to the nearest half, we consider:
- \(6\) is closer than \(7\), and we check the halfway mark, which is 6.5.
- Since \(6.324 < 6.5\), it rounds down to \(6\).
Thus, \(\sqrt{40}\) is between the whole number 6 and rational number 6.0 (also rounded to the nearest half).
Therefore, the answer is:
- Whole number: 6
- Nearest half: 6.0