[tex]\text{The current population is 1.33 billion and the growth rate is 0.49}\%\text{ per year}\\ \\ \text{the exponential model for the population is}\\ \\ P=P_0 e^{rt}\\ \\ \text{here P is the population after t years, }P_0\text{ is the initial population, so }P_0=1.33\\ \text{r is the growth rate, so }r=0.49\%=0.0049\\ \\ \text{now when population will reach, 1.5 billion, we have }P=1.5\\ \\ \text{so using the above formula, we get}[/tex]
[tex]1.5=1.33e^{0.0049t}\\ \\ \text{divide both sides by 1.33, we get}\\ \\ \frac{1.5}{1.33}=e^{0.0049t}\\ \\ \text{take natural log both sides to get}\\ \\ \ln \left ( \frac{1.5}{1.33} \right )=\ln e^{0.0049t}\\ \\ \Rightarrow \ln \left ( \frac{1.5}{1.33} \right )=0.0049t \ln(e)\\ \\ \Rightarrow \ln \left ( \frac{1.5}{1.33} \right )=0.0049t (1)\\ \\ \Rightarrow t=\frac{1}{0.0049}\ln \left ( \frac{1.5}{1.33} \right )\\ \\ \Rightarrow t\approx 24.55[/tex]
Hence the population will reach 1.5 billion after approximately 25 years
