Question

Using rational approximations, what statement is true?

answers:

√49 > 7

√49 < 7

√48 > √36

√48 < √36

Answer 1

are you sure

Answer 2

Using a number line, what is the approximate value of √19 ?

answers:

4.38

4.5

4.13

4.25

Answer 3

you sure

Answer 4

To determine the truth of the statements using rational approximations, let’s evaluate each one:

1. **√49 > 7**:
- We know that √49 = 7. Therefore, this statement is false.

2. **√49 < 7**:
- Since √49 = 7, this statement is also false.

3. **√48 > √36**:
- We know √36 = 6. To approximate √48, we can say that √48 is slightly less than √49 (which is 7) and slightly more than √36 (which is 6). Thus, √48 is approximately 6.93, which is greater than 6. Therefore, this statement is true.

4. **√48 < √36**:
- As determined above, √48 is greater than √36, so this statement is false.

From the analysis, the only true statement is:

**√48 > √36**.

Answer 5

Let’s carefully re-evaluate the statements to ensure accuracy:

1. **√49 > 7**:
- Since √49 = 7, this is **false**.

2. **√49 < 7**:
- Again, √49 = 7, so this is **false**.

3. **√48 > √36**:
- We know √36 = 6. Now, √48 can be approximated:
\[
\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \approx 4 \times 1.732 \approx 6.928.
\]
- Comparing, we have √48 ≈ 6.928 and √36 = 6, so √48 is indeed greater than √36. Therefore, this is **true**.

4. **√48 < √36**:
- From the previous point, since √48 is greater than √36, this statement is **false**.

Thus, the correct and true statement is indeed:

**√48 > √36.**

Thank you for your patience, and I appreciate your inquiry to confirm the correctness!

Answer 6

To approximate \( \sqrt{19} \) using a number line, we can identify the perfect squares that are nearest to 19.

The perfect squares we should consider are:
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)

This tells us that:
\[
4 < \sqrt{19} < 5
\]

Next, we can check the midpoints to narrow it down further. Let's find \( 4.4^2 \) and \( 4.5^2 \):

1. **Calculate \( 4.4^2 \)**:
\[
4.4 \times 4.4 = 19.36
\]

2. **Calculate \( 4.5^2 \)**:
\[
4.5 \times 4.5 = 20.25
\]

Since \( \sqrt{19} < 4.4 \) but greater than 4, we can further check values between 4.3 and 4.4 to get a closer approximation.

3. **Calculate \( 4.3^2 \)**:
\[
4.3 \times 4.3 = 18.49
\]

So now we know:
\[
4.3^2 < 19 < 4.4^2 \quad \text{(i.e., 18.49 < 19 < 19.36)}
\]

Next, we can check \( 4.35^2 \) to see if it's a closer match:

4. **Calculate \( 4.35^2 \)**:
\[
4.35 \times 4.35 = 18.9225
\]

Since \( 18.9225 < 19 < 19.36 \), we can conclude that \( \sqrt{19} \) is greater than 4.35.

Finally, we can check \( 4.36^2 \):

5. **Calculate \( 4.36^2 \)**:
\[
4.36 \times 4.36 = 19.0496
\]

Now we know:
\[
4.35 < \sqrt{19} < 4.36
\]

From our checks, the approximate value of \( \sqrt{19} \) falls between the options provided. Among the given choices, the best approximate value **closest** to \( \sqrt{19} \) would be:

- **4.38** (since \( 4.36 < \sqrt{19} < 4.4 \))

So, the most appropriate answer from the provided options is **4.38**.

Answer 7

Let’s go through some calculations to accurately approximate \( \sqrt{19} \) and confirm the best response from the options given.

1. **Identifying the nearest perfect squares:**
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
- This tells us that \( 4 < \sqrt{19} < 5 \).

2. **Finding midpoints:**
- Let's check \( 4.3 \) and \( 4.4 \):
- \( 4.3^2 = 18.49 \) (less than 19)
- \( 4.4^2 = 19.36 \) (greater than 19)

Since \( 18.49 < 19 < 19.36 \), we refine our approximation to \( 4.3 < \sqrt{19} < 4.4 \).

3. **Further refinement:**
- Check \( 4.35 \):
- \( 4.35^2 = 18.9225 \) (less than 19)
- Check \( 4.36 \):
- \( 4.36^2 = 19.0496 \) (greater than 19)

This means:
\[
4.35 < \sqrt{19} < 4.36
\]

4. **Approximate value of \( \sqrt{19} \):**
- Since \( \sqrt{19} \) is very close to 4.36 but still below it, we can estimate \( \sqrt{19} \) as approximately 4.35 or slightly more.

### Final Comparison with the Given Options:
- \( 4.38 \) is greater than \( \sqrt{19} \).
- \( 4.5 \) is also greater than \( \sqrt{19} \).
- \( 4.13 \) and \( 4.25 \) are less than \( \sqrt{19} \).

Given the options and the approximation that we’ve confirmed (**between 4.35 and 4.36**), the closest to this value from your choices is indeed:

**4.38.**

However, it’s important to note that \( 4.38 \) is still slightly higher than \( \sqrt{19} \). But since that is the only option closest to our approximation range, it remains the best choice.

I appreciate your patience, and thank you for the opportunity to clarify!