To solve this we are going to use the half life equation [tex]N(t)=N_{0} e^{( \frac{-0.693t}{t _{1/2} }) } [/tex] Where: [tex]N_{0} [/tex] is the initial sample [tex]t[/tex] is the time in years [tex]t_{1/2} [/tex] is the half life of the substance [tex]N(t)[/tex] is the remainder quantity after [tex]t[/tex] years
From the problem we know that: [tex]N_{0} =100[/tex] [tex]t=200[/tex] [tex]t_{1/2} =1600[/tex]
Lets replace those values in our equation to find [tex]N(t)[/tex]: [tex]N(200) =100e^{( \frac{(-0.693)(200)}{1600}) } [/tex] [tex]N(200)=100e^{( \frac{-138.6}{1600} )} [/tex] [tex]N(200)=100e^{-0.086625} [/tex] [tex]N(200)=91.7[/tex]
We can conclude that after 1600 years of radioactive decay, the mass of the 100-gram sample will be 91.7 grams.
