To find between which whole number and rational number (to the nearest half) the square root of 105 falls, we first need to calculate the approximate value of √105.
We know that:
- \(10^2 = 100\)
- \(11^2 = 121\)
So, \(10 < \sqrt{105} < 11\).
Next, we approximate √105. We can refine our estimate by calculating:
\[ 10.2^2 = 104.04 \] \[ 10.3^2 = 106.09 \]
Since \(10.2^2\) is less than 105 and \(10.3^2\) is greater than 105, we can conclude:
\[ 10.2 < \sqrt{105} < 10.3 \]
Next, we check if √105 is closer to 10 or to 10.5. Since:
\[ 10.25^2 = 105.0625 \]
and since 105 is closer to 105.0625 than it is to 104.04, we can further narrow this down.
Thus, the square root of 105 is approximately:
\[ \sqrt{105} \approx 10.246 \]
Now let's address the question of rational numbers to the nearest half. The nearest halves to consider around 10.246 are 10.0 and 10.5.
Therefore, we can conclude that:
- The whole number between which √105 lies is 10.
- The nearest half (rational number) is 10.5.
So, summarizing: √105 is between 10 and 10.5.
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