Answer:
12.3 years
Explanation:
The equation of the radioactive decay can be written as follows:
[tex]\frac{N(t)}{N_0}=(\frac{1}{2})^{\frac{t}{\tau_{1/2}}}[/tex] (1)
where
N(t) is the amount of radioactive sample left at time t
N0 is the amount of radioactive sample at time t=0
t is the time passed
[tex]\tau_{1/2}[/tex] is the half-life of the isotope
The problem tells us that after t=36.9 y, the amount of sample which has become stable is 7/8. This means that 7/8 of the sample has already decayed, so the amount of radioactive sample left is
[tex]\frac{N(t)}{N_0}=1-\frac{7}{8}=\frac{1}{8}[/tex]
We can now re-arrange equation (1) by using this information and by substituting t=36.9 y we find:
[tex]\frac{t}{\tau_{1/2}}=log_{1/2} (\frac{N(t))}{N_0})\\\tau_{1/2}=\frac{t}{log_{1/2}(\frac{N(t)}{N_0})}=\frac{36.9 y}{log_{1/2}(1/8)}=\frac{36.9 y}{3}=12.3 y[/tex]
So, the answer is
12.3 years
