Respuesta :
Solution :
Let y = amount of caffeine at a given time in body.
n = number of hours
So [tex]$\frac{dy}{dn}= - \ Y$[/tex]
[tex]$\Rightarrow \frac{dY}{Y}=- 0.01 \ dn[/tex]
ln Y = -0.01 n + C
At n = 0, Y = 130 mg
ln 130 = -0.01 x 0 + C
C = ln 130
[tex]$ \ln \frac{Y}{130} =- 0.01 \ n$[/tex]
[tex]$Y = 130 \ e^{-0.01n}$[/tex]
When Y = 65
[tex]$65 = 130 \ e^{-0.01 n }$[/tex]
[tex]$\frac{65}{130}=e^{-0.01n}$[/tex]
[tex]$\ln \frac{1}{2} = -0.01 \ n$[/tex]
n = 69.315 hours
At n = 24 hours
[tex]$Y = 130 \ e^{-0.01 \times 24}$[/tex]
Y = 102.26 mg
