Let
be the concentration of the chemical at time
weeks after the discovery of the dumping. Let
be the concentration at the time of the dumping, which occurred at some time
. We know that the concentration of the chemical decays according to the equation
for some constant
and the solution of this differential equation is
for some constant
. We know that the concentration has fallen from
at time
to
at time
and to
at time
. Using the latter two pieces of information, we get
and
Dividing equation (1) by equation (2) yields
So, we can find
, where
:
So, based on decay rates, the contamination / dumping occurred 28 weeks and 3 days prior to it being discovered.
Suspect A was imprisoned for 26 weeks ending 1 week prior to the discovery, and so was imprisoned 27 weeks prior to the discovery of the contamination, and so was free at the time the dumping occurred.
Suspect B was imprisoned for 26 weeks ending 13 weeks prior to the discovery, and so was imprisoned at 39 weeks prior to the discovery and was still imprisoned when the dumping occurred.
The contamination decay rates are consistent with Suspect A being free at the time the crime was committed but Suspect B being incarcerated at that time and so unable to commit the crime. Based on these results, Suspect B is innocent and Suspect A remains under suspicion and is potentially the offender.