Question

Using a number line, what whole number and rational number to the nearest half is 68−−√ between? Show your work. (1 point) Responses between 9 and 9.5 between 9 and 9.5 between 8.5 and 9 between 8.5 and 9 between 7.5 and 8 between 7.5 and 8 between 8 and 8.5

Answer 1

To find the value of \(\sqrt{68}\) and identify the interval it falls into, we first compute the square root:

1. Calculate \(\sqrt{68}\):
\[
\sqrt{68} = \sqrt{(4 \times 17)} = \sqrt{4} \times \sqrt{17} = 2\sqrt{17}
\]

2. Next, we need to estimate \(\sqrt{17}\). We know that:
\[
4^2 = 16 < 17 < 25 = 5^2
\]
Therefore, \(4 < \sqrt{17} < 5\).

3. To find a better approximation for \(\sqrt{17}\), we will calculate the squares of numbers between 4 and 5:
- \(4.1^2 = 16.81\)
- \(4.2^2 = 17.64\)

So, \(4.1 < \sqrt{17} < 4.2\).

4. By using this approximation:
\[
2 \times 4.1 < 2\sqrt{17} < 2 \times 4.2
\]
This gives us:
\[
8.2 < \sqrt{68} < 8.4
\]

5. Thus, \(\sqrt{68}\) is approximately between \(8.2\) and \(8.4\).

6. Now, rounding \(\sqrt{68}\) to the nearest half:
- The nearest half values in this range are 8.0, 8.5 (because \(8.2\) is closer to \(8.0\) than \(8.5\)).

Therefore, \(\sqrt{68}\) is between \(8.5\) and \(9\).

The final answer is: **between 8.5 and 9**.