Respuesta :
Answer:
The half-life of the radioactive substance is 135.9 hours.
Step-by-step explanation:
The rate of decay is proportional to the amount of the substance present at time t
This means that the amount of the substance can be modeled by the following differential equation:
[tex]\frac{dQ}{dt} = -rt[/tex]
Which has the following solution:
[tex]Q(t) = Q(0)e^{-rt}[/tex]
In which Q(t) is the amount after t hours, Q(0) is the initial amount and r is the decay rate.
After 6 hours the mass had decreased by 3%.
This means that [tex]Q(6) = (1-0.03)Q(0) = 0.97Q(0)[/tex]. We use this to find r.
[tex]Q(t) = Q(0)e^{-rt}[/tex]
[tex]0.97Q(0) = Q(0)e^{-6r}[/tex]
[tex]e^{-6r} = 0.97[/tex]
[tex]\ln{e^{-6r}} = \ln{0.97}[/tex]
[tex]-6r = \ln{0.97}[/tex]
[tex]r = -\frac{\ln{0.97}}{6}[/tex]
[tex]r = 0.0051[/tex]
So
[tex]Q(t) = Q(0)e^{-0.0051t}[/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^{-0.0051t}[/tex]
[tex]0.5Q(0) = Q(0)e^{-0.0051t}[/tex]
[tex]e^{-0.0051t} = 0.5[/tex]
[tex]\ln{e^{-0.0051t}} = \ln{0.5}[/tex]
[tex]-0.0051t = \ln{0.5}[/tex]
[tex]t = -\frac{\ln{0.5}}{0.0051}[/tex]
[tex]t = 135.9[/tex]
The half-life of the radioactive substance is 135.9 hours.
