Answer:
10-hour decay factor = 0.13012
5-hour decay factor = 0.52173
1-hour decay factor = 0.87799
154.52624 mg of caffeine
Step-by-step explanation:
Exponential decay is the decrease in a quantity N according to:
[tex]N(t) = N_{0} e^{-kt}[/tex]
where
[tex]N_{0}[/tex] = initial value of quantity N
N(t) = quantity N at time t
k = decay constant associated to physical properties of N
[tex]e^{-kt}[/tex] = decay factor
Substituting the values from the problem:
t = 10 hours
[tex]e^{-kt}[/tex] = 0.2722
Then, solving for k:
[tex]0.2722 =e^{-k*10}\\ ln(0.2722)=ln(e^{-k*10})\\ln(0.2722)=-k*10\\\frac{-ln(0.2722)}{10} =k\\0.13012=k[/tex]
With the value of the decay constant k, you could calculate the decay factor at any given time. For t = 5 hours, the decay factor is given by:
[tex]e^{-kt}=e^{-0.13012*5} \\e^{-kt}=0.52173[/tex]
For t = 1 hour, the decay factor is given by:
[tex]e^{-kt}=e^{-0.13012*1} \\e^{-kt}=0.87799[/tex]
If there are 176 mg in Tony's body 1.23 hours after consuming the energy drink, then you could take this value as the initial value of quantity N, i.e. [tex]N_{0}[/tex]. Then, the quantity of caffeine in Tony's body 2.23 hours later is just the quantity N(t) one hour later from the initial value (1.76 mg), then:
[tex]N(t) = N_{0}e^{-kt}\\ N(1) = (176mg)e^{-k*1}\\N(1)=(176mg)(0.87799)\\N(1)=154.52624 mg[/tex]
Note that [tex]e^{-k*1}[/tex] is the 1-hour decay factor previously calculated.
