SOLUTION: suppose a radioactive isotope such that one-seventh of the atoms in a sample decay after four years. Find the half-life of this isotope

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Question 952918: suppose a radioactive isotope such that one-seventh of the atoms in a sample decay after four years. Find the half-life of this isotope
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
Basic Model: highlight_green%28y=p%2Ae%5E%28-kt%29%29

The one-seventh life is 4 years.

Find logarithms of both sides.
ln%28y%29=ln%28p%29-kt
kt=ln%28p%29-ln%28y%29
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You can find two formulas from that step:
k=%28ln%28p%29-ln%28y%29%29%2Ft and t=%28ln%28p%29-ln%28y%29%29%2Fk

FIND THE DECAY CONSTANT k.
k=%28ln%281%29-ln%281%2F7%29%29%2F4

k=%28ln%281%2F%281%2F7%29%29%29%2F4

k=ln%287%29%2F4

FIND THE HALF-LIFE t%5B1%2F2%5D.
t%5B1%2F2%5D=%28ln%28p%2Fy%29%29%2Fk

ln%281%2F%281%2F2%29%29%2F%28ln%287%29%2F4%29

ln%282%29%284%2Fln%287%29%29

highlight%284%28%28ln%282%29%29%2F%28ln%287%29%29%29%29, years