Problem 78 - 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. By \(2009,\) the models indicate that Canada's population will exceed Uganda's by approximately 2.8 million.

Step-by-Step Solution

Answer

Put the calculated difference as the short answer. If the statement is false, provide the correct statement along with the correct difference.

Step 1: Understand the exponential growth models

The models provided for Canada and Uganda are \(A_c=33.1 e^{0.000 t}\) and \(A_u=28.2 e^{0.034 t}\) respectively. Here, \(A_c\) and \(A_u\) represent the populations of Canada and Uganda respectively, \(t\) years after 2006.

Step 2: Find the value of \(t\) for the year 2009

First, find the value of \(t\) for 2009. Since \(t = 0\) indicates the year 2006, \(t\) for 2009 would be \(2009-2006 = 3\).

Step 3: Substitute \(t\) into the equations

Substitute \(t = 3\) into both equations. So for Canada, \(A_c=33.1 e^{0.000 * 3}\) and for Uganda, \(A_u=28.2 e^{0.034 * 3}\).

Step 4: Calculate the numerical values of \(A_c\) and \(A_u\)

Compute numerical value for both \(A_c\) and \(A_u\), this will give you the populations of both Canada and Uganda in 2009 in millions.

Step 5: Subtract the value of \(A_u\) from \(A_c\)

Subtract the value of \(A_u\) from \(A_c\) . If the difference equals 2.8 million, then the given statement is true, otherwise it is false. If the statement is false, identify the correct difference.

Key Concepts

Population Growth

Population growth refers to the increase in the number of individuals in a population over time. In the context of mathematical models, this concept often involves projecting how populations will expand in the future. Population growth can be influenced by factors such as birth rates, death rates, immigration, and emigration.
Exponential growth, specifically, describes a situation where the growth rate of the population is proportional to the current size of the population. This means the larger the population gets, the faster it grows. Such models are very useful in predicting future population sizes and understanding trends over time.
For instance, exponential growth models like the ones used for Canada and Uganda allow us to estimate future population sizes based on current data and growth rates. In this exercise, we're particularly looking at populations in millions and adding time as a variable to see how these figures might change from year to year.

Mathematical Modeling

Mathematical modeling involves creating equations or formulas to represent real-world phenomena. These models help in analyzing situations and making predictions based on current data.
In the exercise, mathematical models for the populations of Canada and Uganda are developed with exponential equations using natural exponential functions. These functions are written in terms such as \[ A = A_0 e^{rt} \] where:

• \( A \) represents the population at time \( t \)
• \( A_0 \) is the initial population
• \( r \) is the constant rate of growth
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828

These models are crucial for understanding potential future scenarios in population dynamics and can inform policy decisions, resource allocation, and planning.

Exponential Equations

Exponential equations are mathematical representations where variables appear in the exponent. This is a key feature of exponential growth models. These equations are fundamental in understanding processes that grow at rates proportional to their size, such as population growth. The general form of an exponential growth equation is:\[ A = A_0 e^{rt} \]
Here, the variable \( t \) represents time, \( A_0 \) is the initial amount, and \( r \) is the growth rate.
Exponential equations are powerful because they can describe situations that change rapidly over time. In our current exercise, they are used to compare population sizes over a specific period and verify whether assumptions or predictions about these populations are accurate.

Algebraic Calculations

Algebraic calculations allow us to solve and manipulate equations to find important pieces of information, like changes in population over time. In our example, we use algebra to plug in the values for \( t = 3 \) (three years after 2006) into the models for Canada and Uganda to compute their populations in 2009.

Let's break this down:

• For Canada: Insert \( t = 3 \) into \( A_c = 33.1 e^{0.000 \, * \, 3} \), simplifying this gives \( A_c = 33.1 \, e^{0} \), which equals 33.1 million.
• For Uganda: Insert \( t = 3 \) into \( A_u = 28.2 e^{0.034 \, * \, 3} \), simplifying this gives \( A_u \approx 30.1 \) million.

Finally, algebraic subtraction provides us with the difference, used to verify the statement about Canada exceeding Uganda's population by 2.8 million in 2009. These calculations can either affirm or correct such assumptions, demonstrating the power of algebra in practical problem-solving.