Question

if your growth rate per year is .49% and the current population is 1.33 billion when will it reach 1.5 billion?

Answer 1

26 years or 26.08 years to be exact

Answer 2

`\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}`

`1.5=1.33e^(0.0049t)\\ \\ \text{divide both sides by 1.33, we get}\\ \\ (1.5)/(1.33)=e^(0.0049t)\\ \\ \text{take natural log both sides to get}\\ \\ \ln \left ( (1.5)/(1.33) \right )=\ln e^(0.0049t)\\ \\ \Rightarrow \ln \left ( (1.5)/(1.33) \right )=0.0049t \ln(e)\\ \\ \Rightarrow \ln \left ( (1.5)/(1.33) \right )=0.0049t (1)\\  \\ \Rightarrow t=(1)/(0.0049)\ln \left ( (1.5)/(1.33) \right )\\ \\ \Rightarrow t\approx 24.55`

Hence the population will reach 1.5 billion after approximately 25 years