Question

2. Using a number line, what whole number and rational number to the nearest half is √59 between? Between 8.5 and 9? 6 and 6.5? 7 and 7.5? 7.5 and 8?

Answer 1

can you use a number line?

Answer 2

To determine where \( \sqrt{59} \) falls on a number line, we can first approximate its value.

Calculating, we find:

\[
7^2 = 49
\]
\[
8^2 = 64
\]

Since \( 49 < 59 < 64 \), we know that:

\[
7 < \sqrt{59} < 8
\]

Next, we can narrow down the value further by calculating:

\[
7.5^2 = 56.25
\]
\[
7.6^2 = 57.76
\]
\[
7.7^2 = 59.29
\]

Since \( 56.25 < 59 < 59.29 \), this means:

\[
7.6 < \sqrt{59} < 7.7
\]

Now, we can approximate it further to see if it lies closer to 7.6 or 7.7.

Let's consider values like \( 7.65 \):

\[
7.65^2 = 58.4225
\]

Since \( 58.4225 < 59 \), we confirm that:

\[
7.65 < \sqrt{59} < 7.7
\]

Thus,

\[
7.6 < \sqrt{59} < 7.7
\]

Now, rounding \( \sqrt{59} \) to the nearest half gives us approximately \( 7.7 \).

To summarize, \( \sqrt{59} \) is found to be between the following ranges:
- **Between 8.5 and 9:** No
- **Between 6 and 6.5:** No
- **Between 7 and 7.5:** No
- **Between 7.5 and 8:** Yes

So the final conclusion:
- The whole number is **7**, and the rational number to the nearest half is **7.5 or 7.7** (however, \( \sqrt{59} \) is indeed closer to 7.7).

Answer 3

Certainly! Let's illustrate the placement of \( \sqrt{59} \) on a number line visually and determine where it falls.

### Step 1: Understand the Value of \( \sqrt{59} \)

From previous calculations, we found that:

\[
7 < \sqrt{59} < 8
\]

To narrow it down further:

- \( 7.5^2 = 56.25 \)
- \( 7.6^2 = 57.76 \)
- \( 7.7^2 = 59.29 \)

So we have:

\[
7.6 < \sqrt{59} < 7.7
\]

### Step 2: Plotting on the Number Line

We can represent this on a number line. Here’s a simplified version of what the number line would look like, highlighting the key points:

```
6 6.5 7 7.5 8 8.5 9
|------|------|------|------|------|------|
^ ^
| |
7.6 7.7
```

### Step 3: Identify the Whole Number and Rational Number to the Nearest Half

- The whole number that \( \sqrt{59} \) is between is **7**.
- The rational number to the nearest half is closer to **7.7**.

### Conclusion

In summary, on a number line:
- **Whole number:** \( 7 \)
- **Rational number to the nearest half:** \( 7.5 \) (but closer to \( 7.7 \)).

So to answer your original question: Yes, \( \sqrt{59} \) is indeed between:

- **7 and 7.5:** **YES**
- **7.5 and 8:** **YES** (but actually closer to 7.7)

It is not between 6 and 6.5, nor between 8.5 and 9.