At midnight, an automatic camera located at an intersection photographs a man dr

Question

At midnight, an automatic camera located at an intersection photographs a man driving erratically enough to suggest that he may be under the influence of alcohol. Two hours later, the police locate the driver and determine there is 0.06% alcohol in his blood. An hour later at police headquarters, a follow-up test detects a blood-alcohol level of 0.05%. Evidence indicates that the suspect consumed no alcohol after midnight. If the legal blood-alcohol level is 0.08%, can the suspect be charged with driving under the influence of alcohol? (Assume that the percentage P of alcohol in the bloodstream satisfies the decay equation P' = -kP.)

Explanation / Answer

P = P0 e^(-kt)
we know at t=2 P =0.06
so
0.06 = P0 e^(-2k)
as well
0.05 = P0 e^(-3k)
divide the two equations
0.06/0.05 = e^k
k = ln(6/5)
so
0.06 = P0 e^( - 2*ln(6/5))
P0=0.0864
so he can be charged

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