The world population was 2860 million people in 1990 and 3370 million people in 2000. Assuming...

Question:

The world population was 2860 million people in 1990 and 3370 million people in 2000. Assuming that the growth rate is proportional to the population size,

i) What is the relative growth k? Also, what is the population growth model?

ii) Estimate the world population in 2033.

iii) When is the world population being 8524 million people?

Population Growth:

We have been told that the population is growing proportional to its size. Keeping this in mind, we can use an exponential growth model. For this growth model, we will be finding the growth rate using the population figures for 1990 and 2000.

i) The growth model we can use is:

$$P(t)=2860e^{kt}$$

Here, t is the number of years since 1990 and k is the relative growth rate.

We now have to find k. As we know that in 2000, i.e. when t=10, the population is 3370 we find it as follows.

\begin{align} &P(10)=2860e^{10k}=3370\\ &10k=\ln \left ( \frac{3370}{2860} \right )\\ &k\approx 0.016409 \end{align}

The completed model will now be:

$$P(t)=2860e^{0.016409t}$$

ii) The world population in 2033, i.e. at t=43, will be:

\begin{align} P(43)&=2860e^{0.016409*43}\\ &=5791.6\,\text{million} \end{align}

iii) The time at which the world population will be 8524 million can be found as follows.

\begin{align} &2860e^{0.016409t}=8524\\ &0.016409t=\ln \left ( \frac{8524}{2860} \right ) \\&t\approx 66.55 \end{align}