Question

Using a number line, what whole number and rational number to the nearest half is 105−−−√
between?

Answer 1

To determine the whole number and the rational number to the nearest half that \(\sqrt{105}\) is between, we first need to calculate \(\sqrt{105}\).

Calculating \(\sqrt{105}\):

1. Find the approximate square root:
- We know that \(10^2 = 100\) and \(11^2 = 121\). Thus, \(\sqrt{105}\) must be between 10 and 11.

2. To get a more precise estimate, we can try the midpoint:
- \(10.2^2 = 104.04\) (which is less than 105)
- \(10.3^2 = 106.09\) (which is more than 105)

This shows that \(10.2 < \sqrt{105} < 10.3\).

For a more fine-tuned approximation, we can compute \(\sqrt{105}\) using a calculator, which gives approximately:

\[
\sqrt{105} \approx 10.246
\]

Now we want to find the whole number and the rational number to the nearest half that \(\sqrt{105}\) is between.

1. The whole number below \(\sqrt{105} \approx 10.246\) is 10.
2. The whole number above is 11.

Now consider the nearest half:
- The nearest half to \(10.246\) is \(10.5\).

In summary:
\(\sqrt{105}\) is between the whole numbers \(10\) and \(11\) and the rational number to the nearest half is \(10.5\).

Thus, we have:

- Whole number: \(10\)
- Rational number to the nearest half: \(10.5\)