Answer:
The age of the earliest known human-like footprints is approximately 3000000 years.
Explanation:
Decay of isotopes is represented by the following ordinary differential equation:
[tex]\frac{dm}{dt} = -\frac{m}{\tau}[/tex] (Eq. 1)
Where:
[tex]\frac{dm}{dt}[/tex] - First derivative of mass with respect to time, measured in miligrams per year.
[tex]m[/tex] - Current mass of the isotope, measured in miligrams.
[tex]\tau[/tex] - Time constant, measured in years.
Now we proceed to obtain the solution of the differential equation:
[tex]\int\limits {\frac{dm}{m} } = -\frac{1}{\tau}\int dt[/tex]
[tex]\ln m = -\frac{t}{\tau}+C[/tex]
[tex]m(t) = e^{-\frac{t}{\tau}+C }[/tex]
[tex]m(t) = m_{o}\cdot e^{-\frac{t}{\tau} }[/tex] (Eq. 2)
Where:
[tex]m_{o}[/tex] - Initial mass of the isotope, measured in miligrams.
[tex]t[/tex] - Time, measured in years.
[tex]\tau[/tex] - Time constant, measured in years.
We proceed to clear time within the formula presented above:
[tex]\ln \frac{m(t)}{m_{o}} = -\frac{t}{\tau}[/tex]
[tex]t = -\tau \cdot \ln \frac{m(t)}{m_{o}}[/tex]
In addition, time constant can be found as a function of half-life:
[tex]\tau = \frac{t_{1/2}}{\ln 2}[/tex] (Eq. 3)
If we know that [tex]t_{1/2} = 500000\,yr[/tex], [tex]m_{o} = 8\,mg[/tex] and [tex]m(t) = 0.125\,mg[/tex], the age of the earliest known human-like footprints is:
[tex]\tau = \frac{500000\,yr}{\ln 2}[/tex]
[tex]\tau \approx 721347.520\,yr[/tex]
[tex]t = -(721347.520\,yr)\cdot \ln \left(\frac{0.125\,mg}{8\,mg} \right)[/tex]
[tex]t \approx 3000000\,yr[/tex]
The age of the earliest known human-like footprints is approximately 3000000 years.
