Problem 79 - Algebra and Trigonometry

Question

The exponential growth models describe the population of the indicated country, \(A,\) in millions, t years after 2006 . $$ \begin{array}{ll} {\text { Canada }} & {A=33.1 e^{0.009 t}} \\ {\text { Uganda }} & {A=28.2 e^{0.034 t}} \end{array} $$ Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The models indicate that in 2013 , Uganda's population will exceed Canada's.

Step-by-Step Solution

Answer

The statement 'The models indicate that in 2013, Uganda's population will exceed Canada's' is true according to the given models.

Step 1: Understand the exponential growth models

The exponential growth models for Canada and Uganda are given respectively as \(A=33.1 e^{0.009 t}\) and \(A=28.2 e^{0.034 t}\), where \(A\) is the population in millions and \(t\) is the number of years after 2006.

Step 2: Calculate the populations for 2013

To calculate the populations for 2013, we need to substitute \(t = 2013 - 2006 = 7\) into both equations. This gives \(A_C = 33.1 e^{0.009 \times 7}\) for Canada and \(A_U = 28.2 e^{0.034 \times 7}\) for Uganda.

Step 3: Evaluate the exponential expressions

Evaluating these expressions using a calculator or similar tool yields \(A_C ≈ 35.16\) million for Canada and \(A_U ≈ 35.88\) million for Uganda.

Step 4: Compare the populations

Comparing these populations, it can be seen that Uganda's population in 2013, as predicted by the models, will be larger than that of Canada.

Key Concepts

Population Growth

Population growth describes the increase in the number of individuals in a population. It can be influenced by factors such as birth rates, death rates, and migration patterns. In the case of Canada and Uganda, the provided exponential growth models quantify this increase over time. These models are significant as they help predict future population sizes and potential needs for resources, planning, and understanding demographic changes.

For instance, an understanding that a country's population is growing rapidly can lead to the implementation of policies addressing urban development, food security, and healthcare services. The models given in our example use the variable 't' to represent years after 2006, which allows us to predict the population at any given time by substituting the value of 't' into the model.

Exponential Function

An exponential function is a mathematical expression of the form \( y = ab^{x} \), where \( b \) is the base and \( x \) is the exponent. In the context of population growth, exponential functions can effectively represent situations where the growth rate is proportional to the current size of the population, leading to rapid increases over time.

In the case of our example, the exponential functions are \( A=33.1e^{0.009t} \) for Canada and \( A=28.2e^{0.034t} \) for Uganda, where \( e \) (approximately 2.71828) is the base of natural logarithms, and the exponent is a product of the growth rate and time in years. The constant factors, 33.1 and 28.2, represent the populations of the respective countries (in millions) at the starting point of 2006, while 0.009 and 0.034 represent the respective annual growth rates. Understanding the shape and behavior of exponential functions is crucial as they start off slowly but can rapidly increase, indicating a population that is growing significantly over time.

Comparing Populations

Comparing populations between different regions or countries involves looking at their sizes, growth rates, and the time period of growth. When using exponential growth models, comparing populations becomes a matter of substituting the relevant values into the respective exponential functions and analyzing the results.

In the textbook exercise, we are tasked with comparing the populations of Canada and Uganda in the year 2013. This is done by calculating the population for each country by substituting \( t=7 \) (since 2013 is 7 years after 2006) into the exponential growth equations provided. The calculations reveal that Uganda's population is predicted to exceed Canada's in 2013, showcasing an application of exponential models in real-world scenarios such as demographics. This ability to compare populations and project future changes is fundamental for governments, organizations, and researchers, as they plan for and respond to population dynamics.