Using an Exponential Function to Model Population Growth

According to the U.S. Census Bureau, the world population reached 6 billion people during 1999 and was growing exponentially. By the middle of 2018, the population had grown to 7.504 billion. The projected world population (in billions of people) t years after 2018, is given by the function $LaTeX: f\left(t\right)=7.504e^{0.00936t}$ $f\left(t\right)=7.504e^{0.00936t}$ .

Based on this model, what will the world population be in 2025?
In what year will the world population reach 9 billion?

Solutions

Since t = 0 represents the year 2018, in 2025, t would be 2025 – 2018 = 7 yr. We must find f(t) when t is 7.
$LaTeX: f\left(t\right)=7.504e^{0.00936t}$ $f\left(t\right)=7.504e^{0.00936t}$
$LaTeX: f\left(7\right)=7.504e^{0.00936\left(7\right)}\approx8.012$ $f\left(7\right)=7.504e^{0.00936\left(7\right)}\approx8.012$
The population will be 8.012 billion in 2025.
$LaTeX: f\left(t\right)=7.504e^{0.00936t}$ $f\left(t\right)=7.504e^{0.00936t}$
For this problem we want f(t) = 9.
$LaTeX: 9=7.504e^{0.00936t}$ $9=7.504e^{0.00936t}$
$LaTeX: \frac{9}{7.504}=e^{0.00936t}$ $\frac{9}{7.504}=e^{0.00936t}$
$LaTeX: \ln\left(\frac{9}{7.504}\right)=\ln e^{0.00936t}=0.00936t$ $\ln\left(\frac{9}{7.504}\right)=\ln e^{0.00936t}=0.00936t$
$LaTeX: t=\frac{\ln\left(\frac{9}{7.504}\right)}{0.00936}\approx19.4$ $t=\frac{\ln\left(\frac{9}{7.504}\right)}{0.00936}\approx19.4$
So 19.4 years after 2018, during the year 2037, world population will reach 9 billion.