Joseph consumes an energy drink that contains caffeine. After consuming the energy drink, the amount of caffeine in Joseph's body decreases exponentia

tricotasu 2020-11-08 Answered
Joseph consumes an energy drink that contains caffeine. After consuming the energy drink, the amount of caffeine in Josephs
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Expert Answer

tabuordy
Answered 2020-11-09 Author has 91 answers

Given,
The 10-hour decay factor for the number of mg of caffeine in Joseph's body is 0.2601.
Exponential decay is the decrease in quantity N according to:
N(t)=N0ekt
where
N0= initial value of quantity N
N(t) = quantity N at time t
k = decay constant associated to physical properties of N
ekt= decay factor
Substituting the values from the problem:
t = 10 hours
ekt=0.2601
Then, solving for k:
Taking log on both side
ln(ekx10)=ln(0.2601)
10k=ln(0.2601)
k=ln(0.2601)10
k =0.13467
Therefore, for 10-hours decay factor = 0.13467
For t = 5 hours, the decay factor is given by:
ekt=e0.13467×5
=e0.67335
= 0.5099
Therefore, for 5-hours decay factor =0.50999
For t = 1 hours, the decay factor is given by:
ekt=e0.13467×1
=e0.13467
=0.87400
Therefore, for 1-hour decay factor =0.87400
If there are 166 mg in Joseph's body 1.38 hours after consuming the energy drink, then you could take this value as the initial value of quantity N0 Then, the quantity of caffeine in Joseph's body 2.38 hours later is just the quantity N(t) one hour later from the initial value (166 mg), then:
N(t)=N0ekt
N(1) =166×0.87400 (since for 1-hour decay factor =0.87400)
N(1)= 145.084 mg

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