Problem 77 - 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. In \(2006,\) Canada's population exceeded Uganda's by 4.9 million.

Step-by-Step Solution

Answer

The statement 'In 2006, Canada's population exceeded Uganda's by 4.9 million' is true.

Step 1: Understanding exponential growth models

It's important to understand the given models. The exponential growth models provided describe the population of Canada and Uganda, \(A(t)\), in millions, \(t\) years after 2006. Therefore, \(A=33.1 e^{0.009 t}\) refers to Canada's population, and \(A=28.2 e^{0.034 t}\) refers to Uganda's population. 't' is the number of years after 2006, and 'A' is the population in millions.

Step 2: Calculate the population in 2006

To validate the statement, first, calculate the population for each country in 2006. Since 2006 is our base year, it means that \(t=0\). For Canada, the population would be \(A=33.1 e^{0.009 \cdot 0}=33.1 millions\). For Uganda, the population would be \(A=28.2 e^{0.034 \cdot 0}=28.2 millions\)

Step 3: Compare the statement with calculations

The statement implies that Canada's population exceeded Uganda's by 4.9 million in 2006. But from the calculations, the difference between Canada's and Uganda's population is \(33.1 - 28.2 = 4.9 millions\)

Step 4: Determine whether the statement is true or false

Based on the calculations, the statement is true as in 2006, Canada's population indeed exceeded Uganda's by 4.9 million. There is no need for any change.

Key Concepts

Population Growth Mathematics

Understanding how populations grow over time is crucial in fields ranging from ecology to economics. The mathematical study of population growth seeks to describe how groups change in size through models. Exponential growth is a specific type of growth pattern where the change in a population's size is proportional to its current size. This means the growth rate accelerates as the population increases.

In the exercise, we see a glimpse into this realm of mathematics with the population models for Canada and Uganda. With the variable 't' representing time in years after 2006, and 'A' the population in millions, we clearly observe that populations don't grow linearly but rather by a constant percentage, depicted by the exponential function's base, 'e'.

Mathematics provides us with tools, like these models, to predict future population sizes, understand demographic changes, and plan for resources and services accordingly.

Exponential Functions

An exponential function is a mathematical expression in which a variable appears in the exponent. In the general form of an exponential function, you’ll see the equation expressed as \( A = a \cdot e^{kt} \) where \( e \) is the base of the natural logarithm, 'a' is the initial amount, 'k' is the growth rate, and 't' is the time.

The beauty of exponential functions lies in their simplicity and power to model complex phenomena such as population growth, radioactive decay, or compound interest. They show how quantities can rapidly increase (or decrease, in the case of exponential decay) over time. For instance, the exercise demonstrates how even a small difference in the growth rate exponent, between Canada's \( 0.009 \) and Uganda's \( 0.034 \), can lead to a significant difference in population dynamics over time.

Comparing Populations

When comparing the populations of two or more regions, it's not just the numbers that matter; the growth rates are pivotal. As demonstrated in the exercise, Canada and Uganda had quite different growth rates. This reflects how population projections can vary significantly based on these rates.

By comparing exponential models of population growth, we can understand and visualize the future demographic changes. We see that even if two countries start with a similar population size, a higher growth rate will lead to a much larger population over time, impacting resource use, urban planning, and socioeconomic policies.

When these comparisons are made accurately, governments can make informed decisions regarding development, healthcare, and infrastructure, ensuring a better quality of life for their inhabitants.