Final answer:
To reduce a sample of radon-222 to 1/16th of its original amount, it would take 4 half-lives, as 1/16th is (1/2)´. Since the half-life of radon-222 is 3.82 days, the total time needed is 4 times 3.82 days, which equals 15.28 days.
Step-by-step explanation:
To calculate how long it will take for only 1/16th of the original sample of radon-222 to remain, we need to understand the concept of half-life. The half-life is the period of time after which half of a sample will have decayed. For radon-222, the half-life is 3.82 days.
Since the question asks for the time when only 1/16th (which is (1/2)´) of the sample remains, we need to calculate how many half-lives it takes for a sample to be reduced to 1/16th of its original amount. This would be:
- After 1 half-life, 1/2 of the original sample remains.
- After 2 half-lives, (1/2)² = 1/4 of the original sample remains.
- After 3 half-lives, (1/2)³ = 1/8 of the original sample remains.
- After 4 half-lives, (1/2)´ = 1/16 of the original sample remains.
Therefore, it takes 4 half-lives to reduce the sample to 1/16th.
To calculate the total time, we multiply the number of half-lives by the half-life duration:
4 half-lives × 3.82 days/half-life = 15.28 days.
After 15.28 days, only 1/16th of the original radon-222 sample remains.