Respuesta :
We need to find a function
[tex]A=A_0e^{kt}[/tex]Suppose that t=0 corresponds to the year 4004BC; then
[tex]\begin{gathered} A(0)=A_0e^{k\cdot0}=A_0e^0=A_0\cdot1=A_0 \\ \Rightarrow A(0)=A_0 \\ \text{and} \\ A(0)=2 \\ \Rightarrow A_0=2 \end{gathered}[/tex]We need to find the value of k. The current year is 2022 and it corresponds to t=4004+2022=6026; then,
[tex]\begin{gathered} 7.9\cdot10^9=A(6026)=2e^{k\cdot6026} \\ \Rightarrow2e^{k\cdot6026}=7.9\cdot10^9 \end{gathered}[/tex]Solving for k,
[tex]\begin{gathered} \Rightarrow e^{6026k}=3.95\cdot10^9 \\ \Rightarrow\ln e^{6026k}=\ln 3.95\cdot10^9 \\ \Rightarrow6026k\ln e=\ln 3.95\cdot10^9 \\ \Rightarrow6026k=\ln 3.95\cdot10^9 \\ \Rightarrow k=\frac{\ln(3.95\cdot10^9)}{6026} \\ \Rightarrow k=0.003666940\ldots \end{gathered}[/tex]Then, the function is
[tex]A(t)=2e^{0.003666940\ldots t}[/tex]Evaluate for the year 2122, this is t=6126
[tex]A(6126)=2e^{0.003666940\ldots\cdot6126}=1.13994\cdot10^{10}[/tex]The population in 2122 will be, approximately, 1.14*10^10 people or 11.4 billion
