To approximate the value of \( \sqrt{103} \) using a number line, we need to find two perfect squares between which 103 lies.
- The perfect square closest to \( 103 \) below it is \( 10^2 = 100 \).
- The perfect square closest to \( 103 \) above it is \( 11^2 = 121 \).
This gives us the inequality:
\[ 10 < \sqrt{103} < 11 \]
Next, we can refine our estimate. Since \( 103 \) is closer to \( 100 \) than to \( 121 \), let's try squaring some numbers between \( 10 \) and \( 11 \).
- Squaring \( 10.1 \):
\[ 10.1^2 = 102.01 \]
- Squaring \( 10.2 \):
\[ 10.2^2 = 104.04 \]
Since \( 102.01 < 103 < 104.04 \), we know:
\[ 10.1 < \sqrt{103} < 10.2 \]
To narrow it down further, let's try:
- Squaring \( 10.15 \):
\[ 10.15^2 = 103.0225 \]
- Squaring \( 10.14 \):
\[ 10.14^2 = 102.6596 \]
Now we have:
\[ 10.14 < \sqrt{103} < 10.15 \]
This suggests that the square root is very close to \( 10.14 \). We can also try something in between:
- Squaring \( 10.141 \):
\[ 10.141^2 \approx 102.851081 \]
- Squaring \( 10.142 \):
\[ 10.142^2 \approx 103.074084 \]
Thus:
\[ 10.141 < \sqrt{103} < 10.142 \]
This shows \( \sqrt{103} \) is approximately \( 10.14 \). Rounding to the nearest hundredth, we find:
\[ \sqrt{103} \approx 10.14 \]
So, the final answer in decimal form, rounded to the nearest hundredth, is:
\[ \boxed{10.14} \]