To find the approximate value of \(\sqrt{103}\) using a number line, we first identify the perfect squares around 103:
- \(\sqrt{100} = 10\)
- \(\sqrt{121} = 11\)
Since \(100 < 103 < 121\), we know that:
\[ 10 < \sqrt{103} < 11 \]
To narrow it down further, we can evaluate the squares of numbers between 10 and 11.
Calculating \(10.1^2\):
\[ 10.1^2 = 102.01 \quad (\text{too low}) \]
Next, we try \(10.2^2\):
\[ 10.2^2 = 104.04 \quad (\text{too high}) \]
Now we know:
\[ 10.1 < \sqrt{103} < 10.2 \]
Next, let’s try \(10.15^2\):
\[ 10.15^2 = 103.0225 \quad (\text{too high}) \]
Now we try \(10.14^2\):
\[ 10.14^2 = 102.8596 \quad (\text{too low}) \]
So, we find that:
\[ 10.14 < \sqrt{103} < 10.15 \]
Next, let’s calibrate it a bit more by trying \(10.145^2\):
\[ 10.145^2 = 103.005025 \quad (\text{too high}) \]
And \(10.143^2\):
\[ 10.143^2 = 102.891249 \quad (\text{too low}) \]
At this point, we can see that \(\sqrt{103}\) is between \(10.143\) and \(10.145\). To get it rounded to the nearest hundredth, we can average these values:
Estimating further, we find that:
\[ \sqrt{103} \approx 10.15 \]
Rounding \(10.144\) to the nearest hundredth gives:
( \sqrt{103} \approx 10.15 )
Thus, the value of \(\sqrt{103}\) rounded to the nearest hundredth is:
\[ \boxed{10.15} \]