A radioactive material is known to decay at a yearly rate proportional to the amount at each moment. There were 2000 grams of the material 10 years ago. There are 1990 grams right now. What is the half-life of the material?
The equation to solve is P = P0e^kt. Where P = population, P0 = initial population, k is a constant and t = time.
To find k:
1990 = p0e^0k
2000 = p0e^-10k
Once I divide these out I get 1990/2000 = e^10k.
To find k take the natural log of both sides:-
ln 1990/2000 = 10k. Therefore k = 1/10*ln 1990/2000.
To solve for the half life:
1000 = 2000e^(1/10 ln 1990/2000)t.
When I work through this I get:-
t = 10* (ln 1/2)/(ln (1990/2000)).
However the correct answer is:-
t = 10* (ln 2)/(ln (2000/1990)).
Much appreciated if someone can show where I am going wrong.