To approximate \( 193\sqrt{5} \) to one decimal place, we first need to find an approximate value for \( \sqrt{5} \).
The square root of 5 can be approximated using a sequence of values:
- Start with a reasonable guess. A common approximation for \( \sqrt{5} \) is \( 2.2 \) because \( 2.2^2 = 4.84 \), which is close to 5.
- Let's refine this value using the average of the guess and \( \frac{5}{\text{guess}} \):
- First iteration: \[ \text{New guess} = \frac{2.2 + \frac{5}{2.2}}{2} = \frac{2.2 + 2.2727}{2} \approx \frac{4.4727}{2} \approx 2.2364 \]
- We can repeat this process:
- Second iteration: \[ \text{New guess} = \frac{2.2364 + \frac{5}{2.2364}}{2} \approx \frac{2.2364 + 2.2361}{2} \approx \frac{4.4725}{2} \approx 2.236067977 \]
- This value is very close to the square root of 5.
However, for practical purposes, we round \( \sqrt{5} \) to approximately \( 2.236 \).
Now, multiplying by 193: \[ 193\sqrt{5} \approx 193 \times 2.236 \approx 431.428 \]
Finally, rounding this to one decimal place: \[ 431.4 \]
Thus, the approximation of \( 193\sqrt{5} \) correct to one decimal place is \( \boxed{431.4} \).
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