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

Question

?(1 point)
Responses

4.5
4.5

4.38
4.38

4.13
4.13

4.25

Using a number line, what whole number and rational number to the nearest half is 105−−−√
between?(1 point)
Responses

between 10.5 and 11
between 10.5 and 11

between 11 and 11.5
between 11 and 11.5

between 10 and 10.5
between 10 and 10.5

between 9.5 and 10

Using rational approximations, what statement is true?(1 point)
Responses

49−−√>7
start root 49 end root greater than 7

49−−√<7
start root 49 end root less than 7

48−−√<36−−√
start root 48 end root less than start root 36 end root

48−−√>36

Using rational approximations, what statement is true?(1 point)
Responses

81−−√>9
start root 81 end root greater than 9

81−−√<9
start root 81 end root less than 9

71−−√>61−−√
start root 71 end root greater than start root 61 end root

71 −−−√<61−−√

Using rational approximations, what statement is true?(1 point)
Responses

12 −−−√<π
start root 12 end root less than pi

16−−√>4
start root 16 end root greater than 4

16 −−−√<4
start root 16 end root less than 4

12 −−−√ >π

Answer 1

Let's evaluate each question one by one:

1. Using a number line, what is the approximate value of \(\sqrt{19}\)?

   The square root of 19 is between the square roots of 16 and 25, which are 4 and 5.

   • \(4.5^2 = 20.25\) (too high)
   • \(4.4^2 = 19.36\) (too high)
   • \(4.3^2 = 18.49\) (too low)

   So \(\sqrt{19} \approx 4.36\).

   The answer is: 4.38 (closest option).

2. What whole number and rational number to the nearest half is \(\sqrt{105}\) between?

   The square root of 105 is between the square roots of 100 and 121, which are 10 and 11.

   • \(10.5^2 = 110.25\) (too high)
   • \(10.4^2 = 108.16\) (too high)
   • \(10.3^2 = 106.09\) (too high)
   • \(10.2^2 = 104.04\) (too low)

   So \(\sqrt{105}\) is between 10.2 and 10.3, and therefore is also between 10.5 and 11.

   The answer is: between 10.5 and 11.

3. Using rational approximations, what statement is true?

   • \(\sqrt{49} = 7\)
   • Therefore, the statement \(\sqrt{49} > 7\) is false.

   The correct statement is: \(\sqrt{49} < 7\) is also false. However, \(\sqrt{48} < \sqrt{36}\) cannot be true since \(48\) is greater than \(36\). So only \(\sqrt{48} < \sqrt{36}\) is true.

   Therefore, no options exactly match the conclusion.

4. Using rational approximations, what statement is true?

   • \(\sqrt{81} = 9\)

   Therefore, the statement \(\sqrt{81} < 9\) is false.

   The correct option is: \(\sqrt{71} < \sqrt{61}\) cannot be true since \(71\) is greater than \(61\). So only \(\sqrt{81} > 9\) is true.

   Therefore, the answer is: \(81 −−−√>9\).

5. Using rational approximations, what statement is true?

   • \(\sqrt{12} \approx 3.46\)
   • \(\pi \approx 3.14\)

   Here, \(\sqrt{12} > \pi\) is true.

   • \(\sqrt{16} = 4\)

   Thus, \(\sqrt{16} > 4\) is also true.

   The answer is: \(16−−√>4\).

So the answers summary would be:

1. 4.38
2. between 10.5 and 11
3. None of the options exactly match but \(\sqrt{48} < \sqrt{36}\) could be understood.
4. \(81−−−√>9\)
5. \(16−−√>4\)