Question

The population (in millions) of a country in 2001 and the expected continuous annual rate of change \(k\) of the population for the years 2000 through 2010 are given. (Source: U.S. Census Bureau, International Data Base) (a) Find the exponential growth model \(P=C e^{k t}\) for the population by letting \(t=0\) correspond to 2000 . (b) Use the model to predict the population of the country in \(2015 .\) (c) Discuss the relationship between the sign of \(k\) and the change in population for the country. $$ \begin{array}{ll} \text { Country } & 2001 \text { Population } & k \\ \end{array} $$ $$ \text { Cambodia } \quad 12.7 \quad 0.018 $$

Short answer

The exponential growth model of Cambodia's population is \(P = 12.7e^{0.018t}\). Using this model, the population can be predicted for any given year. The positive sign of \(k\) (0.018) indicates a continuous annual growth in Cambodia's population.

Step-by-step solution

Find the Exponential Growth Model

The population model is given by \(P = Ce^{kt}\). Given that \(t = 0\) corresponds to the year 2000, for the case of Cambodia the 2001 population (when \(t = 1\)) would be our initial value C. Given data tells us that in 2001, the population was 12.7 million and the constant rate \(k\) is 0.018. Therefore, \(C = 12.7\) and \(k = 0.018\), so the exponential growth model for the population of Cambodia is \(P = 12.7e^{0.018t}\)

Predict the Population in 2015

To predict the population in 2015, replace \(t\) with 15 (since 2000 corresponds to \(t = 0\), 2015 will correspond to \(t = 15\)) in our exponential model. By substituting \(t = 15\), the population \(P\) becomes: \(P = 12.7e^{0.018 \times 15}\). Calculating this will give us the predicted population for Cambodia in 2015.

Discuss the Relationship between k and the Population

The value of \(k\) represents the continuous annual rate of change in the population. If \(k\) is positive, as it is for Cambodia with a value of 0.018, it means that the population is increasing annually at a constant rate. If \(k\) were negative, it would indicate a constant annual decrease in population.