Question

Approximate 193√5
, correct to one decimal place, by using a sequence of values that gets closer and closer to 5–√
.(1 point)

Answer 1

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:

1. 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.
2. Let's refine this value using the average of the guess and \( \frac{5}{\text{guess}} \):
- First iteration:
\[
\text{New guess} = \frac{2.2 + \frac{5}{2.2}}{2} = \frac{2.2 + 2.2727}{2} \approx \frac{4.4727}{2} \approx 2.2364
\]
3. 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 \( \boxed{431.4} \).