Answer:
A. number of decayed atoms = 73.197
Step-by-step explanation:
In order to find the answer we need to use the radioactive decay equation:
where:
N0=initial radioactive atoms
t=time
k=radioactive decay constant
In our case, when t=0 we have 1,000,000 atoms, so:
![1,000,000=N0*e^{k*0}](https://tex.z-dn.net/?f=1%2C000%2C000%3DN0%2Ae%5E%7Bk%2A0%7D)
![1,000,000=N0](https://tex.z-dn.net/?f=1%2C000%2C000%3DN0)
Now we need to find 'k'. Using the provied information that after 365 days we have 973,635 radioactive atoms, we have:
![973,635=1,000,000*e^{k*365}](https://tex.z-dn.net/?f=973%2C635%3D1%2C000%2C000%2Ae%5E%7Bk%2A365%7D)
![ln(973,635/1,000,000)/365=k](https://tex.z-dn.net/?f=ln%28973%2C635%2F1%2C000%2C000%29%2F365%3Dk)
![-0.0000732=k](https://tex.z-dn.net/?f=%20-0.0000732%3Dk)
A. atoms decayed in a day:
![N(t)=1,000,000*e^{-0.0000732t}](https://tex.z-dn.net/?f=N%28t%29%3D1%2C000%2C000%2Ae%5E%7B-0.0000732t%7D)
![N(1)=1,000,000*e^{-0.0000732*1}](https://tex.z-dn.net/?f=N%281%29%3D1%2C000%2C000%2Ae%5E%7B-0.0000732%2A1%7D)
![N(1)= 999,926.803](https://tex.z-dn.net/?f=N%281%29%3D%20999%2C926.803)
Number of atoms decayed in a day = 1,000,000 - 999,926.803 = 73.197
B. Because 'k' represents the probability of decay, then the probability that on a given day 51 radioactive atoms decayed is k=0.0000732.