SOLUTION: A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 28 grams. Write an exponential equation f(t) representing this situatio

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Question 1139499: A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 28 grams. Write an exponential equation
f(t) representing this situation. (Let f be the amount of radioactive substance in grams and t be the time in minutes.)
To the nearest minute, what is the half-life of this substance?

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let n be the number of half-lives. Then, since the substance decays by a factor of 0.5 every half-life, the amount of the original 250g remaining after n half-lives is

f%28t%29+=+250%280.5%29%5En

If the half-life is k minutes, then the number of half-lives in t minutes is t/k. Then the equation is

f%28t%29+=+250%280.5%29%5E%28t%2Fk%29

The half-life k can be determined from the given information that the original 250g decays to 28g in 250 minutes:

28+=+250%280.5%29%5E%28250%2Fk%29
0.112+=+0.5%5E%28250%2Fk%29
log%28%280.112%29%29+=+%28250%2Fk%29%2Alog%28%280.5%29%29
k+=+250%2Alog%28%280.5%29%29%2Flog%28%280.112%29%29+=+79.153 to 3 decimal places.

ANSWER: to the nearest minute, the half-life of the substance is 79 minutes.