To solve this problem step-by-step, let's break down the given information and apply the relevant mathematical concepts.
1. Given Information:
- Initial amount of the sample, [tex]\( A = 100 \% \)[/tex].
- Remaining amount of the sample, [tex]\( P(t) = 18 \% \)[/tex].
- Half-life of Radon-222, [tex]\( \hbar = 3.8 \)[/tex] days.
2. Equation for Decay:
The formula given is:
[tex]\[
P(t) = A \left( \frac{1}{2} \right)^{\frac{t}{\hbar}}
\][/tex]
3. Substitute the known values:
Substitute [tex]\( P(t) = 18 \)[/tex], [tex]\( A = 100 \)[/tex], and [tex]\( \hbar = 3.8 \)[/tex] into the decay formula:
[tex]\[
18 = 100 \left( \frac{1}{2} \right)^{\frac{t}{3.8}}
\][/tex]
4. Simplify the equation:
Divide both sides by 100:
[tex]\[
0.18 = \left( \frac{1}{2} \right)^{\frac{t}{3.8}}
\][/tex]
5. Apply natural logarithm (ln) to both sides:
Taking the natural log of both sides to simplify:
[tex]\[
\ln(0.18) = \ln\left( \left( \frac{1}{2} \right)^{\frac{t}{3.8}} \right)
\][/tex]
Using the log power rule, [tex]\( \ln(a^b) = b \ln(a) \)[/tex]:
[tex]\[
\ln(0.18) = \frac{t}{3.8} \ln\left( \frac{1}{2} \right)
\][/tex]
6. Solve for [tex]\( t \)[/tex]:
Isolate [tex]\( t \)[/tex] by multiplying both sides by 3.8:
[tex]\[
t = \frac{\ln(0.18)}{\ln\left( \frac{1}{2} \right)} \times 3.8
\][/tex]
7. Compute the values:
Using the properties of logarithms and given values:
- [tex]\(\ln(0.18) \approx -1.7148\)[/tex]
- [tex]\(\ln\left( \frac{1}{2} \right) \approx -0.6931\)[/tex]
Hence:
[tex]\[
t = \frac{-1.7148}{-0.6931} \times 3.8 \approx 9.4
\][/tex]
8. Conclusion:
The best estimate for the age of the sample, given that only 18% of the original amount remains and considering the half-life of Radon-222 is 3.8 days, is approximately 9.4 days.
Therefore, the correct answer is [tex]\( \boxed{9.4 \text{ days}} \)[/tex].
