Answer: The footprints were made [tex]2.99\times 10^6years[/tex] years long ago.
Explanation:
Radioactive decay follows first order kinetics.
Half-life of isotope X= 500,000 years
[tex]\lambda =\frac{0.693}{t_{\frac{1}{2}}}=\frac{0.693}{500000}=0.00000139year^{-1}[/tex]
The equation for first order kinetics is :
[tex]N=N_o\times e^{-\lambda t}[/tex]
N = amount left after time t = 0.125 mg
[tex]N_0[/tex] = initial amount = 8.00 mg
[tex]\lambda[/tex] = rate constant
t= time
Putting the values we get:
[tex]0.125=8.00\times e^{- 0.00000139years^{-1}\times t}[/tex]
[tex]0.0156=e^{- 0.00000139years^{-1}\times t}[/tex]
[tex]t=2.99\times 10^6years[/tex]
The footprints were made [tex]2.99\times 10^6years[/tex] years long ago.
