There were 150 mg of a radioactive material in the store at the start of the year 2000. The material has a half life of 15 years . How much material will be there at the start of 2040?

Question

How much material will be there at beginning of 2040?

Answer 1

m=ae^-kt, where m is the mass after t years, a is the initial mass when t=0 and k is the decay constant.

If 15 years is the half-life then when t=15, m=a/2, so a/2=ae^-15k, e^-15k=½.

-15k=-ln(2), k=ln(2)/15, so m=ae^-(t/15)ln(2), which is the same as m=a×2^-t/15.

When a=150mg and t=2040-2000=40, m=150×2^-40/15=150×2^-8/3=23.62mg.

(Quick check: If t=15 years, m would be 150/2=75mg; if t=30 years, m would be 75/2=37.5mg; if t=45 years, m would be 37.5/2=18.75mg. 40 years is between 30 and 45, and 23.62mg is between 18.75mg and 37.5mg.)