SOLUTION
The formula for finding the amount of carbon 14 remaining in time t is given by
[tex]A=A_oa^t[/tex]To get a, we use the information that the half-life is 5730 years. That is in 5730 years,
[tex]A=\frac{A_o}{2}[/tex]Therefore;
[tex]\begin{gathered} \frac{A_o}{2}=A_oa^{5730}_{} \\ canceloutA_O \\ \frac{1}{2}=a^{5730} \\ \text{simplify to get} \\ a=0.999879039 \end{gathered}[/tex]And;
[tex]A=A_o(0.999879039)^t[/tex]b. To find when the object died given that it has 63% of its Carbon 14 remaining today.
[tex]\begin{gathered} 0.63A_0=A_0(0.999879039)^t \\ \text{cancel out A}_0 \\ 0.63=0.999879039^t \\ \text{take the natural log of both sides} \\ \ln 0.63=t(\ln 0.999879039) \\ t=\frac{\ln 0.63}{\ln 0.999879039} \\ t=3819.5 \end{gathered}[/tex]Therefore, the correct answer is option e. 3819.5 years
