To approximate the square root of 103 using a number line, we start by identifying the perfect squares near 103.
- The square of 10 is \(10^2 = 100\).
- The square of 11 is \(11^2 = 121\).
Since 103 is between 100 and 121, we know that:
\[ 10 < \sqrt{103} < 11 \]
Next, we can try to narrow it down further by testing values between 10 and 11:
- Calculate \(10.1^2\): \[ 10.1^2 = 102.01 \]
- Calculate \(10.2^2\): \[ 10.2^2 = 104.04 \]
Now we see that:
\[ 10.1^2 < 103 < 10.2^2 \]
This means:
\[ 10.1 < \sqrt{103} < 10.2 \]
To narrow it down even more, we can test 10.15:
- Calculate \(10.15^2\): \[ 10.15^2 = 103.0225 \]
This tells us:
\[ 10.1 < \sqrt{103} < 10.15 \]
Now, let's check 10.12:
- Calculate \(10.12^2\): \[ 10.12^2 = 102.4144 \]
This gives:
\[ 10.12 < \sqrt{103} < 10.15 \]
Next, we can try 10.13:
- Calculate \(10.13^2\): \[ 10.13^2 = 102.6569 \]
Now we can try 10.14:
- Calculate \(10.14^2\): \[ 10.14^2 = 102.8596 \]
And then 10.141:
- Calculate \(10.141^2\): \[ 10.141^2 \approx 103.000181 \]
Since \(10.141^2\) is slightly greater than 103, we can try \(10.140\):
- Calculate \(10.140^2\): \[ 10.140^2 \approx 102.832\]
Now, moving to \(10.142\):
- Calculate \(10.142^2\): \[ 10.142^2 = 103.000484 \]
At this point, we see:
\[ 10.140 < \sqrt{103} < 10.141 \]
Thus, after approximating, we round \( \sqrt{103} \approx 10.14 \) to the nearest hundredth.
Final answer:
\[ \sqrt{103} \approx 10.14 \]
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