Universal law of radioactive decay states that:
[tex]N=N_0e^{-\lambda t}[/tex]
Where No is an initial number of atoms, N is a number of atoms after time t and lambda is the decay constant. We can use mas of an atom instead of a number of atoms because we know that every atom has the same mass.
[tex]M=M_0e^{-\lambda t}[/tex]
In order to calculate how many atoms are left after 500 days, we need to find decay constant.
We can do this by using the information given in the problem. We know that after 12 days we are left with 55.45 grams and we know the initial amount. With this information we can calculate decay constant:
[tex]M=M_0e^{-\lambda t}\\
55.45=56.8e^{-12\lambda}\\
0.976=e^{-12\lambda}\\
ln(0.976)=-12\lambda\\
-0.0243=-12\lambda\\
\lambda=\frac{0.0243}{12}\\
\lambda=0.002[/tex]
Now we simply plug in t=500 and calculate the answer:
[tex]M=M_0e^{-\lambda t}=56.8 e^{-0.002\cdot 500}=56.8e^{-1}=\frac{56.8}{e}=20.88g[/tex]
The final answer is D (20.88g).
