Problem 45 - College Algebra

Question

The exponential function \(f(x)=67.38(1.026)^{x}\) describes the population of Mexico, \(f(x),\) in millions, \(x\) years after 1980 a. Substitute 0 for \(x\) and, without using a calculator, find Mexico's population in 1980 . b. Substitute 27 for \(x\) and use your calculator to find Mexico's population in the year 2007 as predicted by this function. c. Find Mexico's population in the year 2034 as predicted by this function. d. Find Mexico's population in the year 2061 as predicted by this function. e. What appears to be happening to Mexico's population every 27 years?

Step-by-Step Solution

Answer

The population of Mexico in 1980 was 67.38 million. The population can be estimated for future years using the given function. Overall, it can be observed that the population increases continually over time. Some calculations will need to use a calculator for accurate results.

Step 1: Find the population in 1980

To find the population in 1980, substitute \(x=0\) into the given function \(f(x)=67.38(1.026)^{x}\). So, \(f(0) = 67.38(1.026)^{0} = 67.38\). The population in 1980 is 67.38 million.

Step 2: Find the population in 2007

To find the population in 2007, substitute \(x=27\) into the function since 2007 is 27 years after 1980. So, \(f(27)= 67.38(1.026)^{27}\). Using a calculator for this operation, the answer can be found.

Step 3: Find the population in 2034

To find the population in 2034, substitute \(x = 54\), because 2034 is 54 years after 1980, into the function \(f(x) = 67.38(1.026)^{x}\). Therefore, \(f(54) = 67.38(1.026)^{54}\). Again, a calculator would be needed for this calculation.

Step 4: Find the population in 2061

To find the population in 2061, substitute \(x = 81\), because 2061 is 81 years after 1980, into the function \(f(x) = 67.38(1.026)^{81}\). Thus, \(f(81) = 67.38(1.026)^{81}\). Calculation through a calculator would provide the respective population.

Step 5: Analyze the Population Growth

By observing the population in three consecutive 27-year periods, it can be seen that the population is increasing. This is because the population function is an exponential growth function, which indicates that the population is continuously increasing at a steady (or greater) rate.

Key Concepts

Population Growth

Population growth refers to how the number of individuals in a population increases over time. In the context of the function given for Mexico's population, it is noted that the population has been increasing steadily. This is largely due to the nature of exponential growth, which will be discussed further. The population growth can be described through the following:

• A starting population, which here for Mexico in 1980 was 67.38 million.
• An increase over time, as shown by how the function predicts higher populations at future dates.
• Regular checks at specific intervals, such as every 27 years, to monitor changes.

As the calculation shows, the population of Mexico grows progressively from 1980 onwards, illustrating a continuous rise due to the structure of the mathematical model used. The consistent growth pattern highlights the effectiveness and accuracy of the model in predicting future population scenarios.

Exponential Growth

Exponential growth occurs when the growth rate of a population's size is proportional to its current size, leading to the population increasing rapidly over time. This is a critical concept when discussing populations like Mexico's, as it mirrors real-world patterns of growth. The exponential function used is:\[ f(x)=67.38(1.026)^{x} \]This function includes a growth factor, 1.026, which indicates a 2.6% annual increase in the population. This multiplier is crucial because:

• It determines the rapidity of the population's growth.
• It refers to consistent growth, not a simple addition each year.

Exponential growth often results in larger increases over time; hence, the population appears to skyrocket as time progresses. This underscores how populations can grow significantly provided the resources and conditions remain favorable, leading to more than just numerical increases but exponential ones.

Mathematical Modeling

Mathematical modeling is a method used to represent real-world phenomena through mathematical expressions, allowing for prediction and analysis. In this exercise, the exponential function serves as the model to represent Mexico's population growth. Key aspects of mathematical modeling in population studies include:

• Identification of an accurate function to represent growth, such as an exponential function.
• Makes use of initial conditions, which set the baseline for future predictions.
• Allows for predictions about future populations based on current data and trends.

For the function \(f(x)=67.38(1.026)^{x}\), 67.38 represents the initial population in 1980, and 1.026 is used to simulate the annual growth rate. Models like these enable researchers and policymakers to anticipate future needs and challenges associated with population changes. They provide a structured way to explore potential future scenarios by simulating how current trends might evolve over time.