Using a number line, what is the approximate value of 19−−√?(1 point)

Question

Responses

4.5
4.5

4.13
4.13

4.25
4.25

4.38

Answer 1

To approximate the value of \( \sqrt{19} \) on a number line, we note that:

• \( 4^2 = 16 \)
• \( 5^2 = 25 \)

Since \( 16 < 19 < 25 \), it follows that \( 4 < \sqrt{19} < 5 \).

To get a closer approximation, we can evaluate \( 4.4 \) and \( 4.5 \):

• \( 4.4^2 = 19.36 \) (which is greater than 19)
• \( 4.3^2 = 18.49 \) (which is less than 19)

This places \( \sqrt{19} \) between \( 4.3 \) and \( 4.4 \).

Next, let's check \( 4.35 \):

• \( 4.35^2 = 18.9225 \) (less than 19)

Then \( 4.38 \):

• \( 4.38^2 = 19.1844 \) (greater than 19)

From this, we can see that:

• \( 4.35^2 < 19 < 4.4^2 \)

Thus, \( \sqrt{19} \) is approximately \( 4.35 < \sqrt{19} < 4.4 \).

Given all the options provided, the closest to \( \sqrt{19} \) is:

4.38

Answer 2

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

between 9.5 and 10
between 9.5 and 10

between 10 and 10.5
between 10 and 10.5

between 11 and 11.5
between 11 and 11.5

between 10.5 and 11

Answer 3

To determine where \( \sqrt{105} \) falls on a number line, we can first find the squares of whole numbers that are close to 105.

We know:

• \( 10^2 = 100 \)
• \( 11^2 = 121 \)

Since \( 100 < 105 < 121 \), it follows that:

\[ 10 < \sqrt{105} < 11 \]

Now, we can check the value of \( 10.5 \):

• \( 10.5^2 = 110.25 \) (which is greater than 105)

This means:

• \( \sqrt{105} < 10.5 \)

Next, let's check \( 10.25 \):

• \( 10.25^2 = 105.0625 \) (which is greater than 105)

Now, check \( 10.2 \):

• \( 10.2^2 = 104.04 \) (which is less than 105)

From this, we find that:

\[ 10.2 < \sqrt{105} < 10.25 \]

This provides a more precise range for \( \sqrt{105} \).

Given the options provided, \( \sqrt{105} \) is between:

between 10 and 10.5.

Answer 4

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

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

48−−√>36−−√
start root 48 end root greater than start root 36 end root

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

49−−√<7

Answer 5

To determine which statement is true, let's evaluate each option:

1. \( \sqrt{49} > 7 \)

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

2. \( \sqrt{48} > \sqrt{36} \)

   • \( \sqrt{48} \approx 6.93 \)
   • \( \sqrt{36} = 6 \)
   • Since \( 6.93 > 6 \), the statement \( \sqrt{48} > \sqrt{36} \) is true.

3. \( \sqrt{48} < \sqrt{36} \)

   • As we found above, \( \sqrt{48} \approx 6.93 \) and \( \sqrt{36} = 6 \).
   • Thus, the statement \( \sqrt{48} < \sqrt{36} \) is false.

4. \( \sqrt{49} < 7 \)

   • Since \( \sqrt{49} = 7 \), this statement \( \sqrt{49} < 7 \) is false.

From the above evaluations, the only true statement is:

\( \sqrt{48} > \sqrt{36} \).

Answer 6

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

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

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

71 −−−√<61−−√
start root 71 end root less than start root 61 end root

81−−√<9

Answer 7

Let's evaluate each statement to determine which is true.

1. \( \sqrt{71} > \sqrt{61} \)

   • Since \( 71 > 61 \), we can conclude that \( \sqrt{71} > \sqrt{61} \) is true.

2. \( \sqrt{81} > 9 \)

   • \( \sqrt{81} = 9 \)
   • Therefore, the statement \( \sqrt{81} > 9 \) is false.

3. \( \sqrt{71} < \sqrt{61} \)

   • This is the opposite of the first statement. Since we have already established that \( \sqrt{71} > \sqrt{61} \), this statement is false.

4. \( \sqrt{81} < 9 \)

   • Again, since \( \sqrt{81} = 9 \), the statement \( \sqrt{81} < 9 \) is false.

Based on this evaluation, the true statement is:

\( \sqrt{71} > \sqrt{61} \).

Answer 8

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 less than 4

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

12 −−−√ >π

Answer 9

To evaluate the statements, let's find the approximate values for \( \sqrt{12} \) and \( \pi \), as well as the exact value for \( \sqrt{16} \):

1. Calculate \( \sqrt{12} \):

   • \( \sqrt{12} \approx 3.464 \)

2. \( \pi \):

   • \( \pi \approx 3.14159 \)

3. Calculate \( \sqrt{16} \):

   • \( \sqrt{16} = 4 \)

Now, let's evaluate the statements:

1. \( \sqrt{12} < \pi \)

   • \( 3.464 < 3.14159 \) is false.

2. \( \sqrt{16} < 4 \)

   • \( 4 < 4 \) is false.

3. \( \sqrt{16} > 4 \)

   • \( 4 > 4 \) is false.

4. \( \sqrt{12} > \pi \)

   • \( 3.464 > 3.14159 \) is true.

Based on the evaluations, the true statement is:

\( \sqrt{12} > \pi \).