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

Question

between?
Responses

between 10.5 and 11
between 10.5 and 11 - incorrect

between 10 and 10.5
between 10 and 10.5 - not selected, this is the correct answer

between 9.5 and 10
between 9.5 and 10 - no response given

between 11 and 11.5

Answer 1

To find the square root of 105 (\( \sqrt{105} \)) and determine between which whole number and rational number (to the nearest half) it lies, we can evaluate its approximate value.

1. First, we find the approximate square root of 105. We can estimate √105 by knowing the squares of nearby whole numbers:

   \( 10^2 = 100 \) and \( 11^2 = 121 \).

So, \( \sqrt{105} \) is between 10 and 11.

2. To get a more precise estimate, we can calculate:

   • \( 10.2^2 = 104.04 \)
   • \( 10.3^2 = 106.09 \)

Since \( 104.04 < 105 < 106.09 \), we conclude that \( \sqrt{105} \) is between 10.2 and 10.3.

3. Rounding \( \sqrt{105} \) to the nearest half, we see that it falls between 10 and 10.5.

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

So the correct response to your question is:

• between 10 and 10.5.