Respuesta :
Answer:
The half-life of this substance is of 569.27 days.
Step-by-step explanation:
Amount of a substance after t days:
The amount of a substance after t days is given by:
[tex]P(t) = P(0)e^{-kt}[/tex]
In which P(0) is the initial amount and k is the decay rate, as a decimal.
Suppose a sample of a certain substance decayed to 69.4% of its original amount after 300 days.
This means that [tex]P(300) = 0.694P(0)[/tex]. We use this to find k.
[tex]P(t) = P(0)e^{-kt}[/tex]
[tex]0.694 = P(0)e^{-300k}[/tex]
[tex]e^{-300k} = 0.694[/tex]
[tex]\ln{e^{-300k}} = \ln{0.694}[/tex]
[tex]-300k = \ln{0.694}[/tex]
[tex]k = -\frac{\ln{0.694}}{300}[/tex]
[tex]k = 0.0012[/tex]
So
[tex]P(t) = P(0)e^{-0.0012t}[/tex]
What is the half-life (in days) of this substance?
This is t for which P(t) = 0.5P(0). So
[tex]0.5P(0) = P(0)e^{-0.0012t}[/tex]
[tex]e^{-0.0012t} = 0.5[/tex]
[tex]\ln{e^{-0.0012t}} = \ln{0.5}[/tex]
[tex]-0.0012t = \ln{0.5}[/tex]
[tex]t = -\frac{\ln{0.5}}{0.0012}[/tex]
[tex]t = 569.27[/tex]
The half-life of this substance is of 569.27 days.
