A radioactive substance decays from 60 mg to 10.2 mg in 22 years according to the exponential decay model y=ae-bx, where a is the initial amount and y is the amount remaining after years.

Question

A radioactive substance decays from 60 mg to 10.2 mg in 22 years according to the exponential decay model y=ae^-bx, where a is the initial amount and y is the amount remaining after years.

     Find the b-value.

Use the EXACT b-value to write the exponential decay model for this substance with initial amount  60 mg, then use that model to find the half-life. 

     Find the half-life.

     half life ≈ _____ years

Answer 1

So in this problem y=10.2 a=60 and t = 22. You can substitute these values to give you

10.2=( 60) e^-22b.

.178333= e^-22b Take the natural log of both sides

ln .178333= -22b

b= ln.178333/22 = .078368 so equation is y = 60(e)^.078368t

After one half life there will be 30 gms. left ( half of the original60) so

30 = 60(e)^.078368 t

.5 = e^.078368t take the natural log of both sides

ln .5 = .078368t

t=ln.5/.078368= 8.84years roounded to nearest 1oth