Problem 79 - College Algebra

Question

The exponential growth models describe the population of the indicated country, \(A,\) in millions, \(t\) years after 2006 $$\begin{aligned}&\text { Carada } \quad A=33.1 e^{0.000 t}\\\&\text { Ugaria } \quad A=28.2 e^{0.034 t}\end{aligned}$$ In Exercises \(77-80\), 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 that Uganda's population will exceed Canada's in 2013 is true

Step 1: Identify the given

The following are given: the exponential growth models for Carada \( A=33.1e^{0.000t} \) and for Ugaria \( A=28.2e^{0.034t} \), and the statement to evaluate involves the populations of these countries in 2013 which is 7 years from 2006.

Step 2: Calculate Population for Carada in 2013

Replace the value of \( t \) with 7 in the exponential growth model for Carada to find the population in 2013. Hence, the population of Carada in 2013 becomes \( A=33.1e^{0.000(7)} \). This simplifies to approximately 33.1 million.

Step 3: Calculate Population for Ugaria in 2013

Substitute the value of \( t \) with 7 in the exponential growth model for Ugaria to find the population in 2013. Therefore, the population of Ugaria becomes \( A=28.2e^{0.034(7)} \). Calculating this gives approximately 33.4 million.

Step 4: Compare Populations

We can see from Steps 2 and 3 that the population for Uganda in 2013 does indeed exceed that of Canada’s in the exact year.

Key Concepts

Population Growth

Population growth refers to how the number of individuals in a population increases over time. This concept is vital when considering resource allocation, urban planning, and environmental impact. In mathematical terms, population growth can often be described using exponential models, especially when growth is unrestricted by resources or other limiting factors. In the real world, however, environmental constraints and resource availability typically result in logistic growth, which eventually levels off. In the given exercise, the populations of Carada and Ugaria are modeled exponentially, indicating continual growth over time without immediate limitations. The exercise asks us to predict future population sizes based on current trends using these growth models.

Mathematical Modeling

Mathematical modeling is a powerful tool used to describe real-world phenomena using mathematical expressions. It involves creating equations or models that capture the essential behavior and patterns observed in nature. In the context of population studies, mathematical models can help project future population sizes, understand demographic shifts, and plan for resource needs.

• Identifies trends and patterns in data
• Predicts future scenarios based on current and past data
• Assists in planning and decision-making

In the exercise, exponential functions are used within the mathematical models to represent the population changes in Carada and Ugaria over time. By plugging in specific years such as 2013, the model helps determine the estimated populations for those years, assisting in comparing population dynamics between the two regions.

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent, typically written as \( y = a e^{kt} \), where \( e \) is the base of natural logarithms. These functions are pivotal in describing processes that grow or decay at rates proportional to their current value. Some key characteristics of exponential functions include:

• They model rapid increases (growth) or decreases (decay).
• The growth rate is constant, leading to the function's "J-shaped" curve.
• They are widely used in fields like population studies, finance, and physics.

In the original exercise, exponential functions model population growth for two countries, Carada and Ugaria. The parameters \( 33.1 \) and \( 28.2 \) represent initial population sizes while the growth rates are described by \( k \) values of \( 0.000 \) and \( 0.034 \) respectively. These models allow us to project future population sizes, with Ugaria's higher growth rate hinting at faster population expansion compared to Carada.