Question

A. Plot the pointsB. Add an exponential curve to the graph. The initial population P(0)Is 5099.4, so use P(t) = 5099.4e^rt. Try different values for r, the growth rate, until the curve fits the data well (hint: r is less than .050)C. What is your exponential growth function for the population. D. Compare the population of Madagascar in 2020 with the population your model would predict. Does your model overestimate or underestimate the actual 2020 population?

Answer 1

A. We have to plot the points: the year in the horizontal axis and the population in the vertical axis.

The result is:

B. We now have to add an exponential curve.

We know the initial population, P(0)=5099.4, but we don't know the growth rate "r", so we have to guess its value.

The formula in function of r and year (t) is:

`P(t)=5099.4e^(rt)`

NOTE: the right way is to use linear regression, but it is not required here.

One way to guess r is to take the first and last of the data points: P(0)=5099.4 and P(60)=27691.0.

We then write:

`\begin{gathered} (P(60))/(P(0))=(5099.4\cdot e^(r\cdot60))/(5099.4)=(27691.0)/(5099.4) \\ e^(60r)=(27691)/(5099.4) \\ e^(60r)\approx5.43 \\ \ln (e^(60r))=\ln (5.43) \\ 60r=\ln (5.43) \\ r\approx(\ln(5.43))/(60) \\ r\approx(1.692)/(60) \\ r\approx0.028 \end{gathered}`

If we add the plot of the function with the value of r=0.028, we get:

We can see that the function is very accurate to describe the real population evolution.

C. Then, knowing r=0.028 is a good estimation of the growth rate, we can write the population function as:

`P(t)=5099.4e^(0.028t)`

D. As we use the population of 2020 to estimate the growth rate, we will get an almost exact value when we used the model to predict it. The difference is due to rounding error in this case.

We can calculate it as:

`\begin{gathered} P_(2020)=P(60)=5099.4\cdot e^(0.028\cdot60) \\ P(60)=5099.4\cdot e^(1.68) \\ P(60)=5099.4\cdot5.366 \\ P(60)=27361 \end{gathered}`

As the real population is 27691, we have a difference of 30 underestimating the actual population. This difference is equivalent to 30/27691 = 0.1%.

NOTE: If we had not known the actual population of 2020 and we needed to estimate with data until 2010, we would have estimated r as:

`\begin{gathered} (P(50))/(P(0))=(5099.4\cdot e^(r\cdot50))/(5099.4)=(21151.6)/(5099.4) \\ 50r=\ln ((21151.6)/(5099.4)) \\ r\approx(\ln (4.148))/(50) \\ r\approx(1.423)/(50) \\ r\approx0.028 \end{gathered}`

As the population fits the exponential very well, we get the same result for r (discarding rounding errors).