SOLUTION: a sample of radioactive material decayed to 33% of its original mass after 8 days. find the half life of the material.

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Question 549637: a sample of radioactive material decayed to 33% of its original mass after 8 days. find the half life of the material.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The amount A remaining, after time t is a fraction of the initial amount A%5B0%5D
A%2FA%5B0%5D=%281%2F2%29%5E%28t%2Ft%5B0.5%5D%29, with t%5B0.5%5D being the half-life.
(Note: There are many (equivalent) ways to express that relationship/function, but this is the one that seems to make more direct sense: the fraction is 1%2F2 multiplied times itself as many times as half-lives have passed).
So, taking logarithms of both sides

For the problem, with 33% remaining, A%2FA%5B0%5D=33%2F100=0.33, and t=8 (in days)
So, we can find t%5B0.50%5D%29 (in days), by solving
log%280.33%29=%288%2Ft%5B0.5%5D%29%28-log%282%29%29 --> -log%280.33%29%2Flog%282%29=%288%2Ft%5B0.5%5D%29 --> -log%282%29%2Flog%280.33%29=%28t%5B0.5%5D%2F8%29 --> t%5B0.5%5D=-8%2Alog%282%29%2Flog%280.33%29=5 (rounded)
So the half-life is 5 days.