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

Question

Responses

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
between 11 and 11.5

between 10 and 10.5

Answer 1

To find between which whole numbers and rational numbers to the nearest half \( \sqrt{105} \) falls, we first need to calculate \( \sqrt{105} \).

1. Estimate \( \sqrt{105} \):

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

2. To find a more precise approximation:

   • \( 10.2^2 = 104.04 \)
   • \( 10.3^2 = 106.09 \)
   • Thus, \( \sqrt{105} \) is between \( 10.2 \) and \( 10.3 \).

3. To express \( \sqrt{105} \) to the nearest half:

   • Between \( 10 \) and \( 10.5 \), since \( 10.2 \) and \( 10.3 \) round to \( 10 \).

Overall, \( \sqrt{105} \) is approximately \( 10.246 \), which would round to \( 10.5 \) to the nearest half.

Thus, \( \sqrt{105} \) is between 10 and 10.5.

The correct choice is: between 10 and 10.5.