To find the approximate value of \( \sqrt{103} \) using a number line, we need to find two perfect squares between which 103 lies.
The perfect squares around 103 are:
- \( 10^2 = 100 \)
- \( 11^2 = 121 \)
Thus, we can conclude that: \[ 10 < \sqrt{103} < 11 \]
Next, we can try to get a more accurate approximation by estimating between 10 and 11. Calculate the squares of numbers between 10 and 11:
- \( 10.1^2 = 102.01 \) (which is less than 103)
- \( 10.2^2 = 104.04 \) (which is greater than 103)
Since \( 10.1 < \sqrt{103} < 10.2 \), we can narrow it down further.
Now testing a midpoint:
- \( 10.15^2 = 103.0225 \) (which is greater than 103)
- \( 10.12^2 = 102.4144 \) (which is less than 103)
So now we have \( 10.12 < \sqrt{103} < 10.15 \).
Let's check \( 10.13 \):
- \( 10.13^2 = 102.6569 \) (which is less than 103)
Next, check \( 10.14 \):
- \( 10.14^2 = 102.8596 \) (still less than 103)
Finally, check \( 10.145 \):
- \( 10.145^2 = 102.899025 \) (still less than 103)
Now check \( 10.14 \):
- \( 10.14^2 = 102.8596 \)
- Check \( 10.141 \): \[ 10.141^2 \approx 102.8579 \] (still less)
Now check \( 10.142 \): \[ 10.142^2 \approx 102.8770 \] (still less)
Then finally check \( 10.143 \): \[ 10.143^2 \approx 102.8940 \] (still less)
So we see \( 10.144^2 \): \[ 10.144^2 \approx 102.8910 \]
Now we can conclude that \( \sqrt{103} \) is about 10.14 when rounded to the nearest hundredth. Thus the approximate value of \( \sqrt{103} \) rounded to the nearest hundredth is: \[ \sqrt{103} \approx 10.15 \]
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