Question

Using a number line, what whole number and rational number to the nearest half is the square root of 97 between?

Answer 1

wrong,try again

Answer 2

To find the square root of 97 and identify the whole number and the rational number (to the nearest half) that it lies between, we first need to estimate the square root.

We know that:
- \(9^2 = 81\)
- \(10^2 = 100\)

Since \(97\) is between \(81\) and \(100\), we have:
\[
9 < \sqrt{97} < 10
\]

Now, let's get a more precise estimate for \(\sqrt{97}\). A good approach is to find the square of numbers between \(9\) and \(10\).

Calculating \(9.5^2\):
\[
9.5^2 = 90.25
\]

Calculating \(9.8^2\):
\[
9.8^2 = 96.04
\]

Calculating \(9.9^2\):
\[
9.9^2 = 98.01
\]

From these calculations:
- \(9.8^2 = 96.04 < 97\)
- \(9.9^2 = 98.01 > 97\)

This means:
\[
9.8 < \sqrt{97} < 9.9
\]

To find the nearest half, let’s check the value \(9.85\):
\[
9.85^2 = 97.0225
\]

From this, we can see \(9.85^2 > 97\). Since \(9.8 < \sqrt{97} < 9.85\), we can confirm:
\[
\sqrt{97} \text{ is between } 9.8 \text{ and } 9.85.
\]

Thus, rounding to the nearest half, \(\sqrt{97} \approx 9.8\) (which is precisely half of \(9.8 + 9.5\)) indicates that \(\sqrt{97}\) lies between:

- Whole number: **9**
- Rational number (to the nearest half): **9.5**

In conclusion:
- Whole number: **9**
- Rational number to the nearest half: **9.5**

Answer 3

Let's reevaluate the square root of 97.

1. **Identify Whole Numbers:**
- The whole numbers we are considering are based on the squares of integers.
- We know:
- \(9^2 = 81\)
- \(10^2 = 100\)
- Therefore, since \(81 < 97 < 100\), we know:
\[
9 < \sqrt{97} < 10
\]

2. **Estimate the Square Root:**
- We can narrow down further between 9 and 10.
- Let's calculate the squares of more decimal values:
- Try \(9.8\):
\[
9.8^2 = 96.04
\]
- Try \(9.9\):
\[
9.9^2 = 98.01
\]
- Now we have:
\[
9.8 < \sqrt{97} < 9.9
\]

3. **Finding the Nearest Half:**
- To find the nearest half, we can check \(9.85\):
\[
9.85^2 = 97.0225
\]
- Since \(9.85^2 > 97\), we can see that:
\[
9.8 < \sqrt{97} < 9.85
\]
- Therefore, \(\sqrt{97}\) is between \(9.8\) and \(9.85\).

Thus, rounding to the nearest half:
- The rational number to the nearest half is \(9.5\) and this value is indeed less than \(\sqrt{97}\).
- But for the whole number:

We confirm:
- Whole number: **9**
- Rational number to the nearest half: **10** (not \(9.5\), since \( \sqrt{97} \approx 9.848\), which rounds down to \(9.5\), and is closer to \(10\)).

In summary:
- Whole number: **9**
- Rational number to the nearest half: **10**

Thank you for your patience!