SOLUTION: suppose a radioactive isotope is such that one-ninth of atoms in a sample decay after eight years. Find the half life of this isotope. Round to one decimal place.

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Question 913640: suppose a radioactive isotope is such that one-ninth of atoms in a sample decay after eight years. Find the half life of this isotope. Round to one decimal place.
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
p, amount present at passage of time t, (years)
I, initial amount for any t=0
k, a constant
p=Ie%5E%28kt%29, the general decay model

One-ninth after 8 years.
1%2F9=1%2Ae%5E%28k%2A8%29
ln%281%2F9%29=ln%28e%5E%288k%29%29
8k%2Aln%28e%29=ln%281%2F9%29
k=%28ln%281%2F9%29%29%2F8
highlight_green%28k=-ln%289%29%2F8%29
highlight_green%28k=-0.2747%29.

The question was, find the half-life.
Ie%5E%28-0.2747t%29=p
1%2Ae%5E%28-0.2747t%29=1%2F2
ln%28e%5E%28-0.2747t%29%29=ln%281%2F2%29
-0.2747t%2A1=ln%281%2F2%29
t=-ln%281%2F2%29%2F%280.2747%29
t=ln%282%29%2F0.2747
t=2.5233-----------highlight%28t=2.5%29 when rounded to nearest tenth.
Rounded to ONE decimal place, 2.5 years, the half-life.