We were given the following information:
[tex]\begin{gathered} Half\text{ }Life=1,820years \\ t_0=300mg \end{gathered}[/tex]We will proceed to solve as shown below:
a)
[tex]\begin{gathered} Half\text{ }Life=1,820years \\ t_0=300mg \\ 1,820\text{ years ago: } \\ t=2\cdot t_0=2\cdot300=600mg \\ t=600mg \\ \\ \therefore1,820\text{ years ago, there was 600mg of Rad-226} \end{gathered}[/tex]b)
[tex]\begin{gathered} Half\text{ }Life=1,820years \\ t_0=300mg \\ t_1=150mg\Rightarrow1,820 \\ t_2=75mg\Rightarrow2(1,820) \\ t_3=37.5mg\Rightarrow3(1,820) \\ A(t)=t_0\cdot(\frac{1}{2})^t \\ \\ \therefore A(t)=t_0\cdot(\frac{1}{2})^t \end{gathered}[/tex]c)
[tex]\begin{gathered} A(t)=t_0\cdot(\frac{1}{2})^t \\ 1period=1,820years \\ t=\frac{4,000}{1,820}=2.1978 \\ t=2.1978 \\ A(t)=300\times(\frac{1}{2})^{2.1978} \\ A(t)=65.3909\approx65.39 \\ A(t)=65.439mg \\ \\ \therefore A(t)=65.39mg \end{gathered}[/tex]