Respuesta :
Answer:
The half-life of the radioactive substance is of 3.25 days.
Step-by-step explanation:
The amount of radioactive substance is proportional to the number of counts per minute:
This means that the amount is given by the following differential equation:
[tex]\frac{dQ}{dt} = -kQ[/tex]
In which k is the decay rate.
The solution is:
[tex]Q(t) = Q(0)e^{-kt}[/tex]
In which Q(0) is the initial amount:
8000 counts per minute on a Geiger counter at a certain time
This means that [tex]Q(0) = 8000[/tex]
500 counts per minute 13 days later.
This means that [tex]Q(13) = 500[/tex]. We use this to find k.
[tex]Q(t) = Q(0)e^{-kt}[/tex]
[tex]500 = 8000e^{-13k}[/tex]
[tex]e^{-13k} = \frac{500}{8000}[/tex]
[tex]\ln{e^{-13k}} = \ln{\frac{500}{8000}}[/tex]
[tex]-13k = \ln{\frac{500}{8000}}[/tex]
[tex]k = -\frac{\ln{\frac{500}{8000}}}{13}[/tex]
[tex]k = 0.2133[/tex]
So
[tex]Q(t) = Q(0)e^{-0.2133t}[/tex]
Determine the half-life of the radioactive substance.
This is t for which Q(t) = 0.5Q(0). So
[tex]Q(t) = Q(0)e^{-0.2133t}[/tex]
[tex]0.5Q(0) = Q(0)e^{-0.2133t}[/tex]
[tex]e^{-0.2133t} = 0.5[/tex]
[tex]\ln{e^{-0.2133t}} = \ln{0.5}[/tex]
[tex]-0.2133t = \ln{0.5}[/tex]
[tex]t = -\frac{\ln{0.5}}{0.2133}[/tex]
[tex]t = 3.25[/tex]
The half-life of the radioactive substance is of 3.25 days.
