Question

1) Write an exponential growth model of the form A = A0e^kt to represent the population growth of the world. The initial population was 2 in 4004 BC and is now approximately7.9 billion. Use your model to predict the world’s population 100 years from now.

Answer 1

We need to find a function

`A=A_0e^(kt)`

Suppose that t=0 corresponds to the year 4004BC; then

`\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}`

We need to find the value of k. The current year is 2022 and it corresponds to t=4004+2022=6026; then,

`\begin{gathered} 7.9\cdot10^9=A(6026)=2e^(k\cdot6026) \\ \Rightarrow2e^(k\cdot6026)=7.9\cdot10^9 \end{gathered}`

Solving for k,

`\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=(\ln(3.95\cdot10^9))/(6026) \\ \Rightarrow k=0.003666940\ldots \end{gathered}`

Then, the function is

`A(t)=2e^(0.003666940\ldots t)`

Evaluate for the year 2122, this is t=6126

`A(6126)=2e^(0.003666940\ldots\cdot6126)=1.13994\cdot10^(10)`

The population in 2122 will be, approximately, 1.14*10^10 people or 11.4 billion