Question

7. Using a number line, what whole number and rational number to
the nearest half is √70 between?

A. 7 and 7.5
B. 7.5 and 8
C.8 and 8.5
D. 8.5 and 9

Answer 1

explain

Answer 2

shorten the answer

Answer 3

To determine the approximate value of \( \sqrt{70} \), we can find the whole numbers it is between.

First, let's identify the perfect squares closest to 70. We know that:

- \( 8^2 = 64 \)
- \( 9^2 = 81 \)

This means that \( \sqrt{70} \) lies between \( 8 \) (since \( \sqrt{64} = 8 \)) and \( 9 \) (since \( \sqrt{81} = 9 \)).

Now, to find the approximate value of \( \sqrt{70} \):
- \( 8.3^2 = 68.89 \) (which is less than 70)
- \( 8.4^2 = 70.56 \) (which is more than 70)

So, \( \sqrt{70} \) is between \( 8.3 \) and \( 8.4 \).

To round to the nearest half, we can consider:
- \( 8.3 \) rounds to \( 8.5 \)
- \( 8.4 \) also rounds to \( 8.5 \)

Thus, \( \sqrt{70} \) is roughly \( 8.37 \), which rounds to \( 8.5 \).

Since \( \sqrt{70} \) is between \( 8 \) and \( 8.5 \) when we look at the ranges:

Looking at the multiple-choice options:
A. 7 and 7.5
B. 7.5 and 8
C. 8 and 8.5
D. 8.5 and 9

The correct answer is:
**C. 8 and 8.5**

Answer 4

Sure! Let’s break this down step by step to explain how we determine where \( \sqrt{70} \) falls on a number line and how we arrive at the correct answer.

### Step 1: Identify the Range of Squares
First, we look for perfect squares near 70:

- \( 8^2 = 64 \)
- \( 9^2 = 81 \)

Since \( 64 < 70 < 81 \), we can conclude that:

\[
8 < \sqrt{70} < 9
\]

### Step 2: Finding a Closer Approximation
To get a better estimate of \( \sqrt{70} \), we can try squaring some numbers between 8 and 9:

- **Testing 8.3:**
\[
8.3^2 = 68.89 \quad (\text{this is less than } 70)
\]

- **Testing 8.4:**
\[
8.4^2 = 70.56 \quad (\text{this is greater than } 70)
\]

From these calculations, we see that \( \sqrt{70} \) lies between \( 8.3 \) and \( 8.4 \).

### Step 3: Rounding to the Nearest Half
Next, we look at these floating-point numbers in the context of rounding to the nearest half:

- \( 8.3 \) rounds to \( 8.5 \)
- \( 8.4 \) also rounds to \( 8.5 \)

Thus, we can approximate \( \sqrt{70} \) to be about \( 8.37 \), which rounds to \( 8.5 \).

### Step 4: Determining the Final Interval
Given that \( \sqrt{70} \) is definitely more than \( 8 \) but less than \( 9 \), we can conclude:

\[
8 < \sqrt{70} < 8.5
\]

### Step 5: Matching with Options
Now, let’s compare this to the answer choices provided:

A. 7 and 7.5
B. 7.5 and 8
C. 8 and 8.5
D. 8.5 and 9

The correct interval that contains \( \sqrt{70} \) is **C. 8 and 8.5**.

### Summary
Therefore, from our calculations, we deduce that \( \sqrt{70} \) is approximately \( 8.37 \), which puts it between 8 and 8.5. This explains why the correct answer to where \( \sqrt{70} \) lies on a number line is **C. 8 and 8.5**.

Answer 5

To find where \( \sqrt{70} \) lies on a number line:

1. **Identify Perfect Squares**:
- \( 8^2 = 64 \) and \( 9^2 = 81 \) ⇒ \( 8 < \sqrt{70} < 9 \).

2. **Approximate \( \sqrt{70} \)**:
- Test values between 8 and 9:
- \( 8.3^2 = 68.89 \) (less than 70)
- \( 8.4^2 = 70.56 \) (greater than 70)
- Thus, \( 8.3 < \sqrt{70} < 8.4 \).

3. **Round**:
- Both 8.3 and 8.4 round to \( 8.5 \).

4. **Conclude Interval**:
- Therefore, \( 8 < \sqrt{70} < 8.5 \).

### Answer:
The correct interval is **C. 8 and 8.5**.