Question

Using a number line, what is the approximate number value of the square root of 103? Provide your answer in decimal form, rounded to the nearest hundredth

Answer 1

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:

\[
\sqrt{103} \approx 10.14
\]