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

Question

between?(1 point).
Responses

between 9.5 and 10
between 9.5 and 10

between 10 and 10.5
between 10 and 10.5

between 11 and 11.5
between 11 and 11.5

between 10.5 and 11
between 10.5 and 11

Answer 1

To determine the range between which \( \sqrt{105} \) falls, we can calculate the approximate value of \( \sqrt{105} \).

1. Finding the square roots of perfect squares near 105:
   • \( 10^2 = 100 \)
   • \( 11^2 = 121 \)

This tells us that \( 10 < \sqrt{105} < 11 \).

2. Estimating \( \sqrt{105} \):
   • We know \( 10.2^2 = 104.04 \) and \( 10.3^2 = 106.09 \).
   • This tells us that \( 10.2 < \sqrt{105} < 10.3 \).

Based on this calculation, we can further refine our estimate to find the rational number to the nearest half:

• Since \( 10.2 \) is greater than \( 10 \) and less than \( 10.5 \), we can conclude that \( \sqrt{105} \) is between 10 and 10.5.

Thus, the correct response is:

between 10 and 10.5