Answer:
It will take 10.96 hours for 90% of the lead to decay.
Step-by-step explanation:
The radioactive isotope of lead, Pb-209, decays at a rate proportional to the amount present at time t
This means that the amount can be modeled by the following function:
[tex]A(t) = A(0)e^{-rt}[/tex]
In which A(0) is the initial amount and r is the decay rate.
Has a half-life of 3.3 hours.
This means that [tex]A(3.3) = 0.5A(0)[/tex]. We use this to find r.
[tex]A(t) = A(0)e^{-rt}[/tex]
[tex]0.5A(0) = A(0)e^{-3.3r}[/tex]
[tex]e^{-3.3r} = 0.5[/tex]
[tex]\ln{e^{-3.3r}} = \ln{0.5}[/tex]
[tex]-3.3r = \ln{0.5}[/tex]
[tex]r = - \frac{\ln{0.5}}{3.3}[/tex]
[tex]r = 0.21[/tex]
So
[tex]A(t) = A(0)e^{-0.21t}[/tex]
How long will it take for 90% of the lead to decay?
This is t for which [tex]A(t) = 0.1A(0)[/tex], that is, 100 - 90 = 10% of the initial amount.
[tex]A(t) = A(0)e^{-rt}[/tex]
[tex]0.1A(0) = A(0)e^{-0.21t}[/tex]
[tex]e^{-0.21t} = 0.1[/tex]
[tex]\ln{e^{-0.21t}} = \ln{0.1}[/tex]
[tex]-0.21t = \ln{0.1}[/tex]
[tex]t = -\frac{\ln{0.1}}{0.21}[/tex]
[tex]t = 10.96[/tex]
It will take 10.96 hours for 90% of the lead to decay.
