Question

Using a number line, what whole number and rational number to the nearest half is square root 40 between?

Answer 1

To find the whole number and the rational number to the nearest half that \(\sqrt{40}\) is between, we can start by calculating \(\sqrt{40}\).

We know that:

\[
\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}
\]

Next, we can approximate \(\sqrt{10}\). We know that \(\sqrt{9} = 3\) and \(\sqrt{16} = 4\), so \(\sqrt{10}\) is between 3 and 4. A more precise estimate for \(\sqrt{10}\) is approximately 3.162. Therefore,

\[
\sqrt{40} \approx 2 \times 3.162 \approx 6.324
\]

Now, we round this to the nearest half:

- The nearest half below 6.324 is 6.0.
- The nearest half above 6.324 is 6.5.

Since 6.324 is closer to 6.5 than it is to 6.0, we round it to 6.5.

Next, we identify the whole number between which \(\sqrt{40}\) lies:

- The whole number below 6.324 is 6.
- The whole number above 6.324 is 7.

So, \(\sqrt{40}\) is between the whole number 6 and the rational number 6.5 (to the nearest half).

In conclusion, \(\sqrt{40}\) is between:

- Whole number: **6**
- Rational number (to the nearest half): **6.5**