Question

Consider the number line with the plotted square roots.



Inspect the number line. Which square roots have a difference of about 0.5?

(1 point)
Responses

11−−√
and 19−−√
square root of 11 and square root of 19

19−−√
and 24−−√
square root of 19 and square root of 24

11−−√
and 14−−√
square root of 11 and square root of 14

14−−√
and 19−−√

Answer 1

To determine which square roots have a difference of about 0.5, we can approximate each square root:

1. \(\sqrt{11} \approx 3.32\)
2. \(\sqrt{14} \approx 3.74\)
3. \(\sqrt{19} \approx 4.36\)
4. \(\sqrt{24} \approx 4.90\)

Now we can calculate the differences:

- Difference between \(\sqrt{11}\) and \(\sqrt{19}\):
\[
|\sqrt{19} - \sqrt{11}| \approx |4.36 - 3.32| \approx 1.04 \quad (\text{not close to } 0.5)
\]

- Difference between \(\sqrt{19}\) and \(\sqrt{24}\):
\[
|\sqrt{24} - \sqrt{19}| \approx |4.90 - 4.36| \approx 0.54 \quad (\text{close to } 0.5)
\]

- Difference between \(\sqrt{11}\) and \(\sqrt{14}\):
\[
|\sqrt{14} - \sqrt{11}| \approx |3.74 - 3.32| \approx 0.42 \quad (\text{close to } 0.5)
\]

- Difference between \(\sqrt{14}\) and \(\sqrt{19}\):
\[
|\sqrt{19} - \sqrt{14}| \approx |4.36 - 3.74| \approx 0.62 \quad (\text{not close to } 0.5)
\]

The pairs that have a difference close to 0.5 are:

- \(\sqrt{19}\) and \(\sqrt{24}\) (about 0.54)
- \(\sqrt{11}\) and \(\sqrt{14}\) (about 0.42)

Therefore, the correct responses are both options:

- **11−−√ and 14−−√ (square root of 11 and square root of 14)**
- **19−−√ and 24−−√ (square root of 19 and square root of 24)**

However, if you need to choose only one based on the higher relevance to the threshold of 0.5, **the best response would be:**

**\( \sqrt{19} \) and \( \sqrt{24} \)** (square root of 19 and square root of 24).