Question

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

between 11 and 11.5
between 11 and 11.5

between 10.5 and 11
between 10.5 and 11

between 9.5 and 10
between 9.5 and 10

between 10 and 10.5
between 10 and 10.5 cm

Answer 1

To determine where \( \sqrt{105} \) lies on the number line, we first need to approximate its value.

Calculating \( \sqrt{105} \):

- We know that \( 10^2 = 100 \) and \( 11^2 = 121 \). Therefore, \( \sqrt{105} \) is between 10 and 11.

To find a more precise approximation, we can calculate a few more values:

- \( 10.5^2 = 110.25 \) (which is greater than 105)
- \( 10.2^2 = 104.04 \) (which is less than 105)
- \( 10.3^2 = 106.09 \) (which is greater than 105)

From this, we can conclude that \( \sqrt{105} \) is between 10.2 and 10.3.

We can also see that:

- \( \sqrt{105} \) is approximately 10.24 (considering \( 10.2^2 \) is less than and \( 10.3^2 \) is more than 105).

Now, rounding 10.24 to the nearest half gives us 10.5.

Therefore, \( \sqrt{105} \) lies:

- between 10 and 10.5 when considering the whole number.
- between 10.0 and 10.5 when considering the nearest half rational number.

So the correct response is:

**between 10 and 10.5**.