Exponential Decay Rate

Question

The half-life of caffeine in a healthy adult's bloodstream is 5.7 hours. Suppose James, a healthy adult, drinks a Grande Latte, which contains 150 mg of caffeine, at 4 p.m. (assume, perhaps unrealistically, all of the caffeine absorbed into James bloodstream immediately). Use the exponential decay model kt A(t) = A e 0 − where 𝑡 is the time since 4 p.m. measured in hours and A(t) is the amount of caffeine in James bloodstream at time t.

1. Find the exponential decay rate, 𝑘, for the caffeine in James bloodstream. Answer in percentage.
2. At what time of the night will there be only 50 mg of caffeine in James bloodstream. Answer in time to the nearest minute
3. Make a table for times at 0 3 9 and 12
4. If James wakes up and drinks another Grande Latte at 8am how much caffeine would be in his system at that point

Answer 1

1/2 = e^rt = e^5.7k

ln.5 = 5.7k

k = ln5/5.7

= about -12.16% mg/hr

50 = 150e^-.1216t

1/3 = e^-.1216t

-.1216t = ln(1/3)

t = -ln(1/3)/.1216

=about 9.03 hours until 150 mg decays to 50 mg

4pm + 9.03 = 13.03 = about 1:02 am in the morning

at 8 am t= 16

A = 150e^-.1216(16) = 150e^-1.95 = about 21.44

21.44 +150 = 171.44 mg in his system after the 8am Grande Latte