Question

Using a number line, what whole number and rational number to the nearest half is 105−−−√ between?(1 point) Responses between 10.5 and 11 between 10.5 and 11 between 11 and 11.5 between 11 and 11.5 between 10 and 10.5 between 10 and 10.5 between 9.5 and 10

Answer 1

To determine the whole number and rational number to the nearest half that \(\sqrt{105}\) lies between, we first need to find the approximate value of \(\sqrt{105}\).

Calculating \(\sqrt{105}\):

We know that:
- \(\sqrt{100} = 10\)
- \(\sqrt{121} = 11\)

Since \(105\) is between \(100\) and \(121\), it follows that:
\[
10 < \sqrt{105} < 11
\]

Next, we can estimate \(\sqrt{105}\) more precisely. Testing the midpoint between 10 and 11:
- \(10.5^2 = 110.25\) (too high)
- \(10.25^2 = 105.0625\) (too high)

We can find that:
- \(10.2^2 = 104.04\) (too low)
- \(10.3^2 = 106.09\) (too high)

This gives us a better estimate:
\[
10.2 < \sqrt{105} < 10.3
\]

When rounding to the nearest half, we consider the intervals:
- \(10\)
- \(10.5\)
- \(11\)

Thus, \(\sqrt{105}\) is greater than \(10.2\) and less than \(10.3\), so it rounds to \(10.5\).

In conclusion:
- The whole number that \(\sqrt{105}\) is between is \(10\) and \(11\).
- The rational number to the nearest half that \(\sqrt{105}\) is between is \(10.5\).

Therefore, the correct response is:
**between 10 and 10.5**.