Question

What whole number and rational number to the nearest half is the square root of 97 between?

Answer 1

To find the whole number and rational number to the nearest half that \(\sqrt{97}\) is between, we first need to estimate \(\sqrt{97}\).

We know:

\[
9^2 = 81 \quad \text{and} \quad 10^2 = 100
\]

Since \(97\) is between \(81\) and \(100\), it follows that:

\[
9 < \sqrt{97} < 10
\]

Next, we can check the squares of some decimal values between \(9\) and \(10\) to determine the approximate value of \(\sqrt{97}\).

Calculating the square of \(9.5\):

\[
9.5^2 = 90.25
\]

Calculating the square of \(9.7\):

\[
9.7^2 = 94.09
\]

Calculating the square of \(9.8\):

\[
9.8^2 = 96.04
\]

Calculating the square of \(9.9\):

\[
9.9^2 = 98.01
\]

Now we can see:

\[
9.8^2 < 97 < 9.9^2
\]

So:

\[
9.8 < \sqrt{97} < 9.9
\]

To round \(\sqrt{97}\) to the nearest half, we check the value 9.5 and 10. Since \(9.8\) is closer to \(10\) than it is to \(9.5\), we approximate \(\sqrt{97}\) to \(9.5\).

Thus, \(\sqrt{97}\) is between the whole number \(9\) and the rational number \(9.5\) when rounded to the nearest half.

In summary, the whole number is \(9\) and the rational number to the nearest half is \(9.5\).