To estimate the value of \( \sqrt{19} \) using a number line, we first need to identify perfect squares that are close to 19. The two perfect squares surrounding 19 are:
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
Since \( 19 \) is between \( 16 \) and \( 25 \), we know that:
\[ 4 < \sqrt{19} < 5 \]
Step 1: Determine Bounds
From our perfect squares:
- \( \sqrt{16} = 4 \)
- \( \sqrt{25} = 5 \)
Therefore, we now refine our estimation to determine where \( \sqrt{19} \) lies between 4 and 5.
Step 2: Estimate Between 4 and 5
Next, we can test some fractions between 4 and 5 to make a closer approximation:
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Testing \( 4.5 \): \[ 4.5^2 = 20.25 \quad \text{(too high)} \]
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Testing \( 4.4 \): \[ 4.4^2 = 19.36 \quad \text{(too high)} \]
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Testing \( 4.3 \): \[ 4.3^2 = 18.49 \quad \text{(too low)} \]
From this, we see that \( 4.3 < \sqrt{19} < 4.4 \).
Step 3: Narrowing Down Further
Now we can try testing a value between 4.3 and 4.4:
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Testing \( 4.35 \): \[ 4.35^2 = 18.9225 \quad \text{(too low)} \]
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Testing \( 4.38 \): \[ 4.38^2 = 19.1844 \quad \text{(too high)} \]
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Testing \( 4.37 \): \[ 4.37^2 = 19.0569 \quad \text{(too high)} \]
Step 4: Final Estimation
Since \( 4.36 \) is not tested yet: \[ 4.36^2 = 19.0496 \quad \text{(too high)} \]
Now, \( 4.34 \): \[ 4.34^2 = 18.8356 \quad \text{(too low)} \]
Conclusion
From our analysis, we see that:
\[ 4.34 < \sqrt{19} < 4.38 < 4.4 \]
With \( 4.37 \) being too high and \( 4.36 \) being too high, we can conclusively say:
The approximate value of \( \sqrt{19} \) is closest to 4.38 on the number line. Thus, the correct response is:
4.38.