Question

Using a number line, what whole number and rational number to the nearest half is square root 58 between?

Answer 1

To find the whole number and the rational number to the nearest half that \(\sqrt{58}\) falls between, we can first calculate the approximate value of \(\sqrt{58}\).

Calculating \(\sqrt{58}\):

- We know that \(7^2 = 49\) and \(8^2 = 64\).
- Therefore, \(\sqrt{58}\) is between 7 and 8.

For a more precise approximation of \(\sqrt{58}\):
- We can try \(7.5\):
\[
7.5^2 = 56.25 \quad (\text{too low})
\]
- Now try \(7.6\):
\[
7.6^2 = 57.76 \quad (\text{still too low})
\]
- Now try \(7.7\):
\[
7.7^2 = 59.29 \quad (\text{too high})
\]

Since \(7.6^2\) is \(57.76\) (less than \(58\)) and \(7.7^2\) is \(59.29\) (greater than \(58\)), we conclude that:
\[
7.6 < \sqrt{58} < 7.7
\]

Now, rounding to the nearest half, \(7.6\) rounds up to \(7.5\) and \(7.7\) rounds down to \(7.5\) as well. Thus, \(\sqrt{58} \approx 7.62\) falls between the following values:

- **Whole Number**: The whole number it is between is \(7\) (since it's less than \(8\)).
- **Rational Number to the Nearest Half**: The rational number to the nearest half it falls between is \(7.5\).

In summary:
- Whole number: **7**
- Rational number (to the nearest half): **7.5**