When studying radioactive material, a nuclear engineer found that over 365 days, 1,000,000 radioactive atoms decayed to 973 comma 635 radioactive atoms, so 26 comma 365 atoms decayed during 365 days. a. Find the mean number of radioactive atoms that decayed in a day. b. Find the probability that on a given day, 51 radioactive atoms decayed.
Accepted Solution
A:
Answer:A. number of decayed atoms = 73.197Step-by-step explanation:In order to find the answer we need to use the radioactive decay equation:[tex]N(t)=N0*e^{kt}[/tex] where:N0=initial radioactive atomst=timek=radioactive decay constantIn our case, when t=0 we have 1,000,000 atoms, so:[tex]1,000,000=N0*e^{k*0}[/tex][tex]1,000,000=N0[/tex]Now we need to find 'k'. Using the provied information that after 365 days we have 973,635 radioactive atoms, we have:[tex]973,635=1,000,000*e^{k*365}[/tex][tex]ln(973,635/1,000,000)/365=k[/tex][tex] -0.0000732=k[/tex]A. atoms decayed in a day:[tex]N(t)=1,000,000*e^{-0.0000732t}[/tex][tex]N(1)=1,000,000*e^{-0.0000732*1}[/tex][tex]N(1)= 999,926.803[/tex]Number of atoms decayed in a day = 1,000,000 - 999,926.803 = 73.197B. Because 'k' represents the probability of decay, then the probability that on a given day 51 radioactive atoms decayed is k=0.0000732.