Question

Which rational number is closest to the square root of 30?

Answer 1

To find the rational number that is closest to the square root of 30, we can start by estimating the value of \(\sqrt{30}\).

Calculating the squares of some integer values, we have:

\[
5^2 = 25 \quad \text{and} \quad 6^2 = 36
\]

Since \(25 < 30 < 36\), we know that:

\[
5 < \sqrt{30} < 6
\]

Next, we can find a better estimate by trying some decimal values.

Calculating \(5.5\):

\[
5.5^2 = 30.25
\]

Since \(30.25\) is greater than \(30\), we know that:

\[
5 < \sqrt{30} < 5.5
\]

Now, we’ll try \(5.4\):

\[
5.4^2 = 29.16
\]

Since \(29.16\) is less than \(30\), we know:

\[
5.4 < \sqrt{30} < 5.5
\]

Next, we’ll try \(5.48\):

\[
5.48^2 = 30.3004
\]

Now since \(30.3004\) is greater than \(30\), we check \(5.47\):

\[
5.47^2 = 29.9009
\]

Now we have:

\[
5.47 < \sqrt{30} < 5.48
\]

This indicates \(\sqrt{30}\) is approximately \(5.477\).

The two closest rational numbers around \(5.477\) would be \(5.5\) and \(5.4\). Comparing their distance from \(\sqrt{30}\):

- \(5.4\) is \( |5.4 - \sqrt{30}| \approx |5.4 - 5.477| = 0.077\)
- \(5.5\) is \( |5.5 - \sqrt{30}| \approx |5.5 - 5.477| = 0.023\)

Since \(5.5\) is closer to \(5.477\) than \(5.4\), the rational number closest to \(\sqrt{30}\) is:

\[
\boxed{5.5}
\]