Answer:
The material is 30 years old.
Explanation:
The decay of radioactive isotopes can be modelled by the following ordinary differential equation:
[tex]\frac{dm}{dt} = -\frac{m(t)}{\tau}[/tex] (Eq. 1)
Where:
[tex]m(t)[/tex] - Current mass of the isotope, measured in grams.
[tex]\tau[/tex] - Time constant, measured in years.
[tex]\frac{dm}{dt}[/tex] - Rate of change of the isotope in time, measured in grams per year.
After some handling, we find that the solution of the differential equation is:
[tex]\frac{m(t)}{m_{o}} = e^{-\frac{t}{\tau} }[/tex] (Eq. 2)
Where:
[tex]m_{o}[/tex] - Initial mass of the isotope, measured in grams.
[tex]t[/tex] - Time, measured in years.
Now we clear time within the expression above:
[tex]t = -\tau\cdot \ln \frac{m(t)}{m_{o}}[/tex]
Besides, we determine the time constant in terms of the half-life ([tex]t_{1/2}[/tex]), measured in years:
[tex]\tau = \frac{t_{1/2}}{\ln 2}[/tex] (Eq. 3)
If we know that [tex]t_{1/2} = 10\,yr[/tex] and [tex]\frac{m(t)}{m_{o}} = 0.125[/tex], then the age of the material is:
[tex]\tau = \frac{10\,yr}{\ln 2}[/tex]
[tex]\tau = 14.427\,yr[/tex]
[tex]t = -(14.427\,yr)\cdot \ln 0.125[/tex]
[tex]t \approx 30\,yr[/tex]
The material is 30 years old.
