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

4.25

4.5

4.38

4.13

5 answers

To estimate the value of \(\sqrt{19}\) using a number line, we can start by identifying the perfect squares close to 19:

  • \(4^2 = 16\)
  • \(5^2 = 25\)

This means that \(\sqrt{19}\) is between 4 and 5.

Now, let's consider the squares of numbers between 4 and 5 to narrow it down further:

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

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

Since 19 is closer to 18.49 than to 19.36, we can also check \(4.35\):

  • \(4.35^2 = 18.9225\) (which is less than 19)
  • \(4.37^2 = 19.0969\) (which is greater than 19)

At this point, we can see that \(\sqrt{19}\) is approximately between \(4.35\) and \(4.37\).

Now comparing it to the options you provided:

  • 4.25
  • 4.5
  • 4.38
  • 4.13

The option \(4.38\) is actually the closest approximation to \(\sqrt{19}\).

Therefore, the approximate value of \(\sqrt{19}\) is:

4.38

Using a number line, what whole number and rational number to the nearest half is √105 between?

between 10 and 10.5

between 11 and 11.5

between 9.5 and 10

between 10.5 and 11

To determine where \(\sqrt{105}\) lies on the number line and identify a whole number and a rational number to the nearest half, we first find the perfect squares closest to 105:

  • \(10^2 = 100\)
  • \(11^2 = 121\)

This indicates that \(\sqrt{105}\) is between 10 and 11.

Now, let's check the values roughly:

  • \(10.5^2 = 110.25\) (which is greater than 105)
  • \(10.4^2 = 108.16\) (which is also greater than 105)
  • \(10.3^2 = 106.09\) (which is greater than 105)
  • \(10.2^2 = 104.04\) (which is less than 105)

From this analysis, we can conclude:

\(\sqrt{105}\) is between \(10.2\) and \(10.3\), which means it is closer to 10.2 and 10.3 but does not yet exceed 10.5.

Therefore, \(\sqrt{105}\) lies between:

  • Whole number: 10
  • Rational number to the nearest half: 10.5

Thus, the answer is:

between 10 and 10.5.

Using rational approximations, what statement is true?

√48> √36

√48< √36

√49< 7

√49 > 7

To analyze the statements given, let's first calculate the square roots where necessary:

  1. Calculating \(\sqrt{48}\) and \(\sqrt{36}\):

    • \(\sqrt{36} = 6\)
    • \(\sqrt{48}\) can be approximated. Since \(48\) is between \(36\) and \(49\) (where \(\sqrt{49} = 7\)), we can find:
      • \(6^2 = 36\)
      • \(7^2 = 49\)

    To find a more precise estimate of \(\sqrt{48}\), we can note that \(48\) is closer to \(49\) than \(36\), thus: \[ \sqrt{48} \approx \sqrt{36 + 12} \quad (\text{Using the fact that it's between } 6 \text{ and } 7) \] We can also calculate \(\sqrt{48} \approx 6.93\) to get a more precise approximation.

    Therefore:

    • \(\sqrt{48} \approx 6.93 \quad (<7)\)
    • \(\sqrt{36} = 6\)

    So, we have \(\sqrt{48} > \sqrt{36}\) or \(6.93 > 6\).

  2. Calculating \(\sqrt{49}\):

    • \(\sqrt{49} = 7\)

Now, let's evaluate the statements provided:

  • \(\sqrt{48} > \sqrt{36}\): This is true since \(6.93 > 6\).

  • \(\sqrt{48} < \sqrt{36}\): This is false.

  • \(\sqrt{49} < 7\): This is false since \(\sqrt{49} = 7\).

  • \(\sqrt{49} > 7\): This is also false since \(\sqrt{49} = 7\).

Conclusion:

The true statement is:

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