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

Question

Responses

between 10 and 10.5
between 10 and 10.5

between 9.5 and 10
between 9.5 and 10

between 10.5 and 11
between 10.5 and 11

between 11 and 11.5

Answer 1

To determine where \( \sqrt{105} \) lies on a number line, we can calculate its approximate value.

Calculating \( \sqrt{105} \):

1. We know that \( 10^2 = 100 \) and \( 11^2 = 121 \). So, \( \sqrt{105} \) must be between 10 and 11.
2. To narrow it down, we can try some values:
   • \( 10.2^2 = 104.04 \) (too low)
   • \( 10.3^2 = 106.09 \) (too high)

This tells us that \( \sqrt{105} \) is between 10.2 and 10.3.

To find the nearest half (0.5), we note that:

• 10.2 rounds to 10.0
• 10.3 rounds to 10.5

Thus, \( \sqrt{105} \) is between approximately 10.2 and 10.3, so it is:

between 10 and 10.5.

So the correct response is: between 10 and 10.5.