An unknown radioactive element decays into non-radioactive substances. In 440 days the radioactivity of a sample decreases by 74 percent.(a) What is the half-life of the element?half-life: (days)(b) How long will it take for a sample of 100 mg to decay to 91 mg?time needed: (days)

Answer:a) The half life of the element is 231 days.b) It is going to take around 31.5 days for a sample of 100 mg to decay to 91 mg.Step-by-step explanation:The radioactivity of the sample can be modeled by the following exponential equation:[tex]R(t) = R(0)e^{-rt}[/tex]In which t is the time in days, r is the decay rate and [tex]R(0)[/tex] is the initial radioactive percentage.We have that:In 440 days the radioactivity of a sample decreases by 74 percent.This means that [tex]R(440) = 0.26R(0)[/tex].This helps us find r.[tex]R(t) = R(0)e^{-rt}[/tex][tex]0.26R(0) = R(0)e^{-440r}[/tex][tex]e^{-440r} = 0.26[/tex]Applying ln to both sides.[tex]\ln{e^{-440r}} = \ln{0.26}[/tex][tex]-440r = -1.347[/tex][tex]r = 0.003[/tex](a) What is the half-life of the element?This is t when [tex]R(t) = 0.50R(0)[/tex][tex]R(t) = R(0)e^{-rt}[/tex][tex]0.50R(0) = R(0)e^{-0.003t}[/tex][tex]e^{-0.003t} = 0.5[/tex]Again, we apply ln to both sides of the equality.[tex]\ln{e^{-0.003t}} = \ln{0.5}[/tex][tex]-0.003t = -0.693[/tex][tex]t = 231[/tex]The half life of the element is 231 days.(b) How long will it take for a sample of 100 mg to decay to 91 mg?This is t when [tex]R(t) = 0.91R(0)[/tex][tex]R(t) = R(0)e^{-rt}[/tex][tex]0.91R(0) = R(0)e^{-0.003t}[/tex][tex]e^{-0.003t} = 0.91[/tex][tex]\ln{e^{-0.003t}} = \ln{0.91}[/tex][tex]-0.003t = -0.09[/tex][tex]t = 31.44[/tex]It is going to take around 31.5 days for a sample of 100 mg to decay to 91 mg.