The radioactive isotope of lead, Pb-209, decays at a rate proportional to the amount present at time t and has a half-life of 3.3 hours. If 1 gram of this isotope is present initially, how long will it take for 90% of the lead to decay? (Round your answer to two decimal places.)

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.

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