Question

Using a number line, what whole number and rational number to the nearest half is 97−−√ between?(1 point)
97−−√ is between

Answer 1

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

Calculating \( \sqrt{97} \):

1. We know that \( 9^2 = 81 \) and \( 10^2 = 100 \). Thus, \( \sqrt{97} \) is between 9 and 10.
2. To get a more precise value, we can estimate. Since \( 97 \) is closer to \( 100 \) than to \( 81 \), we estimate that \( \sqrt{97} \) is closer to \( 10 \).

Calculating further:
- By trying \( 9.8 \):
\( 9.8^2 = 96.04 \) (too low)
- Trying \( 9.9 \):
\( 9.9^2 = 98.01 \) (too high)

That's enough to conclude that:
\[
9.8 < \sqrt{97} < 9.9
\]

To express \( \sqrt{97} \) to the nearest half:
- The nearest halves around 9.8 are 9.5 and 10.
- Since \( \sqrt{97} \) is approximately 9.8, it rounds to **10**.

Combining this information, we find:
- The whole number is **9** (the largest whole number less than \( \sqrt{97} \)).
- The nearest rational number to half is **10**.

Therefore, \( \sqrt{97} \) is between:
**9 and 10.**