To determine the whole number and rational number (to the nearest half) that the square root of 40 falls between, we start by calculating the square root of 40.
The square root of 40 can be simplified as follows: \[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}. \]
Next, we approximate \(\sqrt{10}\). Since we know: \[ 3^2 = 9 \quad \text{and} \quad 4^2 = 16, \] we see that \(\sqrt{10}\) is between 3 and 4. A better approximation is \(\sqrt{10} \approx 3.16\) (since \(3.16^2 \approx 10\)).
Now we can approximate \(\sqrt{40\): \[ \sqrt{40} \approx 2 \times 3.16 \approx 6.32. \]
Now, we determine the whole numbers around 6.32:
- The whole number below 6.32 is 6.
- The whole number above 6.32 is 7.
Next, we find rational numbers to the nearest half. Since 6.32 is closer to 6.5 than to 6.0, we round it to the nearest half:
- The nearest half to 6.32 is 6.5.
Therefore, the whole number that \(\sqrt{40}\) is between is 6, and the nearest half is 6.5.
In summary, \(\sqrt{40}\) is between:
- Whole number: 6
- Rational number (nearest half): 6.5