To estimate \( \sqrt{329} \) correct to two decimal places, we can use the approximation of \( \sqrt{29} \).
First, let's calculate \( 329 \) in terms of \( 29 \):
\[ \sqrt{329} = \sqrt{29 \times 11.344827586} \approx \sqrt{29} \times \sqrt{11.344827586} \]
Now we can estimate \( \sqrt{11.344827586} \). A rough approximation is \( 3.37 \) (since \( 3.37^2 \approx 11.34 \)).
Hence: \[ \sqrt{329} \approx \sqrt{29} \times 3.37 \]
Now, the value of \( \sqrt{29} \) is approximately \( 5.385 \).
Using this approximation: \[ \sqrt{329} \approx 5.385 \times 3.37 \approx 18.16 \]
However, since we need to find \( \sqrt{329} \) and check which approximation for \( \sqrt{29} \) can accurately give us \( \sqrt{329} \) within two decimal accuracy.
Now let’s analyze the options:
- 5.385: This is a good approximation for \( \sqrt{29} \).
- 5.3851648: This offers more precision than needed for two decimal places.
- 371.00: This cannot be correct, as it’s an order of magnitude off.
- 5.38516: This has more detail than the first option but is still more than what is needed.
The option with the least number of decimal places that sufficiently correlates to two decimal place accuracy in estimating \( \sqrt{329} \) is:
5.385