In exponential decay, a half-life is the same length of time each time the amount remaining is cut in half. Track the amount through successive halvings until it reaches the target, then multiply the number of halvings by the years per half-life. Don’t subtract the same number of grams each time: each loss comes from a smaller remaining amount.
Hints
- Hint 1
A half-life halves the amount still in the sample. After the first half-life, what mass remains from the original grams?
- Hint 2
Keep halving the new amount, not the starting amount. Count how many half-lives it takes to reach grams, then turn that count into years.
Step-by-step
Count the half-lives
Step 1Find the mass after one half-life
No Desmos needed. The halving pattern gives the number of periods directly. A half-life is the time it takes for the amount remaining to shrink by half. Starting with grams, one half-life leaves grams.
- Step 2
Count the halvings needed to reach the target
Keep halving the amount left: . Those are two more half-lives, or three in all from grams. Each half-life halves what remains, not the original amount.
- Step 3
Convert half-lives to years
Each of the three half-lives lasts about years. Multiply the number of half-lives by the years per half-life:
So the sample reaches grams after approximately 17,190 years. Choice C.