Compare 42−−√ and 58−−√ plotted on the number line. What is the approximate difference in tenths between the two values? (1 point) Responses 58−−√ is about 1.2 less than 42−−√ . square root of 58 is about 1.2 less than square root of 42. 58−−√ is about 1.8 greater than 42−−√ . square root of 58 is about 1.8 greater than square root of 42. 58−−√ is about 0.8 greater than 42−−√ . square root of 58 is about 0.8 greater than square root of 42. 58−−√ is about 1.2 greater than 42−−√ . square root of 58 is about 1.2 greater than square root of 42.

Answer 1

To compare \(\sqrt{42}\) and \(\sqrt{58}\), we can find approximate values for both square roots.

1. For \(\sqrt{42}\):

   • We know that \(6^2 = 36\) and \(7^2 = 49\), so \(\sqrt{42}\) is between 6 and 7.
   • A more accurate approximation can be made: \(\sqrt{42} \approx 6.48\) (this can be calculated or referenced).

2. For \(\sqrt{58}\):

   • We know that \(7^2 = 49\) and \(8^2 = 64\), so \(\sqrt{58}\) is between 7 and 8.
   • A more accurate approximation can be made: \(\sqrt{58} \approx 7.62\).

Now, to find the difference between \(\sqrt{58}\) and \(\sqrt{42}\):

\[ \sqrt{58} - \sqrt{42} \approx 7.62 - 6.48 = 1.14. \]

This result indicates that \(\sqrt{58}\) is approximately 1.14 greater than \(\sqrt{42}\).

Rounded to the nearest tenth, the difference is about 1.1.

Among the options given, the closest response is:

"58−−√ is about 1.2 greater than 42−−√. square root of 58 is about 1.2 greater than square root of 42."

Answer 2

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 14−−√ square root of 11 and square root of 14 14−−√ and 19−−√ square root of 14 and square root of 19 11−−√ and 19−−√ square root of 11 and square root of 19 19−−√ and 24−−√ square root of 19 and square root of 24

Answer 3

To determine which pairs of square roots have a difference of about 0.5, we can approximate the square roots for each pair:

1. \(\sqrt{11}\) and \(\sqrt{14}\):

   • \(\sqrt{11} \approx 3.32\)
   • \(\sqrt{14} \approx 3.74\)
   • Difference: \(3.74 - 3.32 \approx 0.42\)

2. \(\sqrt{14}\) and \(\sqrt{19}\):

   • \(\sqrt{14} \approx 3.74\)
   • \(\sqrt{19} \approx 4.36\)
   • Difference: \(4.36 - 3.74 \approx 0.62\)

3. \(\sqrt{11}\) and \(\sqrt{19}\):

   • \(\sqrt{11} \approx 3.32\)
   • \(\sqrt{19} \approx 4.36\)
   • Difference: \(4.36 - 3.32 \approx 1.04\)

4. \(\sqrt{19}\) and \(\sqrt{24}\):

   • \(\sqrt{19} \approx 4.36\)
   • \(\sqrt{24} \approx 4.90\)
   • Difference: \(4.90 - 4.36 \approx 0.54\)

Now, looking at the calculated differences, the pairs that yield a difference close to 0.5 are:

• \(\sqrt{11}\) and \(\sqrt{14}\) (approximately 0.42)
• \(\sqrt{19}\) and \(\sqrt{24}\) (approximately 0.54)

Since both pairs are reasonably close to 0.5, but the requirement is for only one option, the pair that most closely approaches 0.5 is:

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