My Notes OAsk Your Tea 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 85% of the lead to decay? (Round your answel to two decimal places.) hr

Answer:10.96 hours will take for 85% of the lead to decay.Step-by-step explanation:Suppose A represents the amount of Pb-209 at time t,According to the question,[tex]\frac{dA}{dt}\propto A[/tex][tex]\implies \frac{dA}{dt}=kA[/tex][tex]\int \frac{dA}{A}=\int kdt[/tex][tex]ln|A|=kt+C_1[/tex][tex]A=e^{kt+C_1}[/tex][tex]A=e^{C_1} e^{kt}[/tex][tex]\implies A=C e^{kt}[/tex]Let [tex]A_0[/tex] be the initial amount,[tex]A_0=C e^{0} = C[/tex][tex]\implies A=A_0 e^{kt}[/tex]Since, the half-life of 3.3 hours.[tex]\implies \frac{A_0}{2}=A_0 e^{3.3k}\implies e^{3.3k}=0.5\implies k=-0.21004[/tex][tex]\implies A=A_0 e^{-0.21004t}[/tex]Here, [tex]A_0=1\text{ gram}[/tex][tex]A=(100-85)\% \text{ of }A_0=15\%\text{ of }A_0=0.15A_0[/tex]By substituting the values,[tex]0.15A_0=A_0 e^{-0.21004t}[/tex][tex]0.15=e^{-0.21004t}[/tex][tex]\implies t\approx 10.96\text{ hour}[/tex]