Using an exponential function, it is found that it will take 824 minutes for the amount of medication in Aldi’s bloodstream to be effectively 0.
What is an exponential function?
A decaying exponential function for a substance amount is modeled by:
[tex]A(t) = A(0)e^{-kt}[/tex]
In which:
- A(0) is the initial value.
- k is the decay rate, as a decimal.
In this problem, the actual half-life of amoxicillin is about 62 minutes, hence:
[tex]A(62) = 0.5A(0)[/tex]
This is used to find k.
[tex]A(62) = 0.5A(0)[/tex]
[tex]0.5A(0) = A(0)e^{-62k}[/tex]
[tex]e^{-62k} = 0.5[/tex]
[tex]\ln{e^{-62k}} = \ln{0.5}[/tex]
[tex]-62k = \ln{0.5}[/tex]
[tex]k = -\frac{\ln{0.5}}{62}[/tex]
[tex]k = 0.01117979323[/tex]
Hence, the equation is:
[tex]A(t) = A(0)e^{-0.01117979323t}[/tex]
The amount is effectively 0 when [tex]A(t) \approx 0 = 0.0001A(0)[/tex], hence:
[tex]0.0001A(0) = A(0)e^{-0.01117979323t}[/tex]
[tex]e^{-0.01117979323t} = 0.0001[/tex]
[tex]\ln{e^{-0.01117979323t}} = \ln{0.0001}[/tex]
[tex]-0.01117979323t = \ln{0.0001}[/tex]
[tex]t = -\frac{\ln{0.0001}}{0.01117979323}[/tex]
[tex]t = 824[/tex]
It will take 824 minutes for the amount of medication in Aldi’s bloodstream to be effectively 0.
You can learn more about exponential functions at https://brainly.com/question/25537936
