Answer:
[tex]A=0.0122\ gr[/tex]
Step-by-step explanation:
The exponential decrease formula that we must use is:
[tex]A=A_0e^ {-kt}[/tex]
Where A is the amount of thorium as a function of time
[tex]A_0[/tex] is the initial amount and k is the decrease rate
We know that initially there are 50 gr and that you decay 50% every 1.9 years.
So
[tex]A=A_0*0.5=50*0.5 = 25\\\\t=1.9[/tex]
[tex]25=50e^ {-k(1.9)}[/tex]
We solve the equation for k
[tex]0.5=e^ {-k(1.9)}[/tex]
[tex]ln(0.5)=ln(e^ {-k(1.9)})[/tex]
[tex]ln(0.5)=-k(1.9)[/tex]
[tex]-1.9k=ln(0.5)[/tex]
[tex]k=-\frac{ln(0.5)}{1.9}\\\\k=0.3648[/tex]
After 22.8 years the amount of thorium that exists is:
[tex]A=50e^ {-0.3648(22.8)}[/tex]
[tex]A=0.0122\ gr[/tex]
