\begin{frame} \frametitle{Exponential Population Growth} \begin{exampleblock}{} The world population was \begin{itemize} \item 2560 million in 1950, and \item 3040 million in 1960. \end{itemize} Assume a constant growth rate. Find a formula $P(t)$ with \begin{itemize} \item $P(t)$ in millions of people and \item $t$ in years since 1950. \end{itemize} \pause We have \begin{talign} &P(t) = P(0) e^{kt} \\ &\mpause[1]{P(0) = 2560}\\ &\mpause[2]{P(10) = 2560 e^{10k} = 3040}\\ &\mpause[3]{e^{10k} = \frac{3040}{2560}} \mpause[4]{\;\;\implies\;\; k = \frac{1}{10}\ln \frac{3040}{2560} \approx 0.017} \end{talign} \pause\pause\pause\pause\pause The world population growths with a rate of $1.7\%$ per year. \end{exampleblock} \end{frame}