An unknown radioactive element decays into non-radioactive substances. In 30 days the radioactivity of a sample decreases by 12%. When will a sample of 50 mg decay to 10 mg? Round your final answer to 1 decimal place.

Accepted Solution

Answer:Time to decay will be 377.7 days.Step-by-step explanation:Decay of an radioactive element is represented by the formula[tex]A_{t}=A_{0}e^{-kt}[/tex]where [tex]A_{t}[/tex] = Amount after t days[tex]A_{0}[/tex] = Initial amountt = duration for the decayk = decay constantNow we plug in the values in the formula[tex](1-0.12)x=xe^{-30k}[/tex][tex](0.88)x=xe^{-30k}[/tex][tex]0.88=e^{-30k}[/tex]Now we take natural log on both the sides ln(0.88) = [tex]ln(e)^{-30k}[/tex]ln(0.88) = -30k(lne)-30k = -0.1278k = [tex]\frac{.1278}{30}[/tex]k = [tex]4.261\times 10^{-3}[/tex]Now we have to calculate the duration for the decay of 50 mg sample to 10 mg.[tex]A_{t}=A_{0}e^{-kt}[/tex]We plug in the values in the formula10 = 50[tex]e^{-4.261\times 10^{-3}\times t}[/tex][tex]e^{-4.261\times 10^{-3}\times t}=\frac{10}{50}[/tex][tex]e^{-4.261\times 10^{-3}\times t}=0.2[/tex]We take (ln) on both the sides[tex]ln(e^{-4.261\times 10^{-3}\times t})=ln(0.2)[/tex][tex]-4.261\times 10^{-3}\times t=-1.6094[/tex]t = [tex]\frac{1.6094}{4.261\times 10^{-3} }[/tex]t = 0.37771×10³t = 377.7 daysTherefore, time for decay will be 377.7 days.