To solve the problem of determining how much of a 300 gram sample of radon-222 remains after 14.8 days, given its half-life is 3.8 days, we need to follow these steps:
1. Determine the number of half-lives that have elapsed:
- The half-life of radon-222 is 3.8 days.
- The total elapsed time we are considering is 14.8 days.
- To find out how many half-lives have passed, we divide the total elapsed time by the half-life duration.
[tex]\[
\text{Number of half-lives} = \frac{\text{elapsed time}}{\text{half-life duration}} = \frac{14.8 \text{ days}}{3.8 \text{ days}} \approx 3.895
\][/tex]
2. Calculate the remaining mass:
- The decay process follows an exponential decay model, where the remaining mass is calculated as:
[tex]\[
\text{Remaining mass} = \text{initial mass} \times (0.5)^{\text{number of half-lives}}
\][/tex]
- Plug in the values:
[tex]\[
\text{Remaining mass} = 300 \text{ grams} \times (0.5)^{3.895}
\][/tex]
- Evaluating the exponent:
[tex]\[
(0.5)^{3.895} \approx 0.067
\][/tex]
- Then, applying this value:
[tex]\[
\text{Remaining mass} = 300 \text{ grams} \times 0.067 \approx 20.2 \text{ grams}
\][/tex]
Therefore, after 14.8 days, approximately 20.2 grams of the original 300 gram radon-222 sample remains. This matches the closest option from the given choices, which is approximately 20.2 grams.
