SOLUTION: xgrammes of radioactive is isotope decayed to 5g in 100 days.the half life of the isolate is 25 days.calculate mass x

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Question 772377: xgrammes of radioactive is isotope decayed to 5g in 100 days.the half life of the isolate is 25 days.calculate mass x
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
What's probably expected is blind application of a memorized formula involving e while using a calculator and talking about exponential functions and exponential decay.

THE FANCY WAY:
Exponential decay functions state that the amount A remaining after a time t is related to the initial amount A%5B0%5D and the half-life t%5B%221%2F2%22%5D} by the function
ln%28A%29=ln%28A%5B0%5D%29-%28ln%282%29%2Ft%5B%221%2F2%22%5D%29%2At or the corresponding exponential A=A%5B0%5D%2Ae%5E%28-%28t%2Aln%282%29%2Ft%5B%221%2F2%22%5D%29%29
Substituting the values given,
ln%285%29=ln%28x%29-%28ln%282%29%2F25%29%2A100-->ln%285%29=ln%28x%29-ln%282%29%2A%28100%2F25%29-->ln%285%29=ln%28x%29-4%2Aln%282%29-->ln%28x%29=ln%285%29%2B4%2Aln%282%29
ln%28x%29=ln%285%29%2B4%2Aln%282%29%7D%7D--%3E%7B%7B%7Bln%28x%29=ln%285%29%2Bln%282%5E4%29-->ln%28x%29=ln%285%29%2Bln%2816%29-->ln%28x%29=ln%2816%2A5%29-->ln%28x%29=ln%2816%2A5%29--> highlight%28x=80%29

THE CONCEPTUAL, MENTAL MATH WAY:
However, 100 days is 100%2F25=4 half lives, and that means the amount x was halves 4 times, and is now 1%2F16 of the original amount.
Doubling the final 5g amount 4 times, gives us the original amount as
highlight%28x=80g%29
The calculation can be done without touching pencil or calculator.
5 --> 10 --> 20 --> 40 --> 80