Radon, `._(86)^(222)Rn`, is a radioactive gas that can be trapped in the basement of homes, and its presence in high concnetrations is a known health hazard. Radon has a half-life of `3.83 days`. A gas sample contains `4.0 xx 10^(8)` radon atoms initially.
(a) How many atoms will remain after `12 days` have passed if no more radon leaks in? (b) What is the initial activity of the radon sample?

Explanation

To solve the problem step by step, we will break it down into two parts: (a) calculating the number of radon atoms remaining after 12 days and (b) finding the initial activity of the radon sample.

### Part (a): Number of Atoms Remaining After 12 Days

1. **Identify the given values**:
- Initial number of radon atoms, \( n_0 = 4.0 \times 10^8 \) atoms.
- Half-life of radon, \( t_{1/2} = 3.83 \) days.
- Time elapsed, \( t = 12 \) days.
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