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

Question

Responses

4.38
4.38

4.25
4.25

4.5
4.5

4.13
4.13
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Answer 1

To find the approximate value of \(\sqrt{19}\), we can consider perfect squares that are close to 19. The perfect squares are:

• \(4^2 = 16\)
• \(5^2 = 25\)

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

To refine our estimate, we can check values between 4 and 5:

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

So, \(\sqrt{19}\) is between 4.3 and 4.4.

Let's check \(4.35\):

• \(4.35^2 = 18.9225\) (still too low)

Now, let's check \(4.36\):

• \(4.36^2 = 19.0496\) (too high)

Then we check \(4.34\):

• \(4.34^2 = 18.8356\) (too low)

After some calculations, we can conclude that \(\sqrt{19}\) is approximately \(4.36\), which is close to \(4.38\).

So the closest answer from your options would be:

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 10 and 10.5
between 10 and 10.5

between 9.5 and 10
between 9.5 and 10

between 10.5 and 11
between 10.5 and 11

between 11 and 11.5

Answer 3

To determine the value of \(\sqrt{105}\) and its position on a number line, we can examine the perfect squares:

• \(10^2 = 100\)
• \(11^2 = 121\)

Since \(100 < 105 < 121\), we can conclude that:

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

Next, we can refine the estimate of \(\sqrt{105}\) by checking values between 10 and 11:

• \(10.2^2 = 104.04\) (too low)
• \(10.3^2 = 106.09\) (too high)

This shows \(\sqrt{105}\) is between \(10.2\) and \(10.3\).

Now, let's round to the nearest half:

• The nearest halves around \(10.2\) and \(10.3\) are \(10\) and \(10.5\).

Thus, we can conclude that:

\(\sqrt{105}\) is between 10 and 10.5.