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

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

14−−√
and 19−−√
square root of 14 and square root of 19

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

11−−√
and 19−−√

Answer 1

To determine which square roots have a difference of about 0.5, let's first approximate the square roots of the given numbers:

• \( \sqrt{11} \approx 3.32 \)
• \( \sqrt{14} \approx 3.74 \)
• \( \sqrt{19} \approx 4.36 \)
• \( \sqrt{24} \approx 4.90 \)

Now we can calculate the differences between these square roots:

1. \( \sqrt{19} - \sqrt{24} \approx 4.36 - 4.90 = -0.54 \) (this is about 0.5)

2. \( \sqrt{14} - \sqrt{19} \approx 3.74 - 4.36 = -0.62 \) (this is about 0.5)

3. \( \sqrt{11} - \sqrt{14} \approx 3.32 - 3.74 = -0.42 \) (this is about 0.5)

4. \( \sqrt{11} - \sqrt{19} \approx 3.32 - 4.36 = -1.04 \) (this is not close to 0.5)

Now let's summarize the approximate differences:

• \( \sqrt{19} \) and \( \sqrt{24} \) have a difference of approximately 0.54.
• \( \sqrt{14} \) and \( \sqrt{19} \) have a difference of approximately 0.62.
• \( \sqrt{11} \) and \( \sqrt{14} \) have a difference of approximately 0.42.

Thus, the pairs that have differences close to 0.5 are:

• \( \sqrt{19} \) and \( \sqrt{24} \)
• \( \sqrt{14} \) and \( \sqrt{19} \)
• \( \sqrt{11} \) and \( \sqrt{14} \)

From the options provided, the pairs that have a difference of about 0.5 are:

• \( \sqrt{19} \) and \( \sqrt{24} \) (1. option)

So the correct answer is: \( \sqrt{19} \) and \( \sqrt{24} \).