For this problem, we use the integrated rate law for first order radioactive decay which is expressed as follows:
An = Aoe^-kt
where An is the amount left after time t, Ao is
the initial amount and k is a constant.
We need to calculate first the value of k from the given ration An/Ao and the time it reached that ration. We do as follows:
An = Aoe^-kt
0.46= e^-k(9)
k = 0.0863 / day
At half life, the remaining substance would be equal to one-half of the original so that An/Ao would be equal to 1/2 or 0.50. We calculate the half-life as follows:
An = Aoe^-kt
0.50 = e^-0.0863(t)
t = 8.03 days
