consider the number line with the plotted square roots.

Question

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

14−−√
and 19−−√
square root of 14 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

Answer 1

To find which square roots have a difference of about 0.5, we can calculate the approximate values of the square roots involved:

• \( \sqrt{11} \approx 3.3166 \)
• \( \sqrt{14} \approx 3.7417 \)
• \( \sqrt{19} \approx 4.3589 \)
• \( \sqrt{24} \approx 4.8989 \)

Now, we can calculate the differences between the square roots:

1. \( |\sqrt{11} - \sqrt{14}| \approx |3.3166 - 3.7417| \approx |0.4251| \) (about 0.43)
2. \( |\sqrt{14} - \sqrt{19}| \approx |3.7417 - 4.3589| \approx |0.6172| \) (about 0.62)
3. \( |\sqrt{19} - \sqrt{24}| \approx |4.3589 - 4.8989| \approx |0.54| \) (about 0.54)
4. \( |\sqrt{11} - \sqrt{19}| \approx |3.3166 - 4.3589| \approx |1.0423| \) (about 1.04)

Based on the approximations above, the two square roots that have a difference of about 0.5 are:

\( \sqrt{19} \) and \( \sqrt{24} \), with a difference of about 0.54.