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Population modeling is a powerful tool used to understand how populations change over time. It helps predict future population sizes based on current data. One common type of population modeling is exponential growth, where a population increases by a consistent percentage each year.
In the case of Mexico, we start with a set initial population in 1980 of 67.38 million. By using the exponential growth formula, we can model this population's change over time with the expectation that it grows by a simple rate, consistently over the years. The formula that represents this scenario is:

• \( P(t) = P_0 \cdot (1 + r)^t \)

Here, \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is time passed since the starting year, 1980. This formula provides a framework to calculate population sizes at any point in the future, assuming no change in the growth rate. In the example, the model enables us to calculate yearly populations and predict milestones like when the population will reach a certain size, such as 90 million. Thus, population modeling offers insights and helps planners make informed decisions about future resource needs and policy development.
By understanding how populations evolve, we gain valuable foresight into potential challenges and opportunities related to demographics.

Functional notation provides a convenient and precise way of expressing mathematical functions that describe various relationships, like population growth. It uses a straightforward format to denote the operation applied to a set of numbers. In our context, we used it to calculate the population at any given year since the initial point.

• Let's specifically look at how to write the population function of Mexico in 1983. We have already defined our formula for exponential growth as \( P(t) = 67.38 \cdot (1.026)^t \), where \(P(t)\) is the population in millions at time \(t\) years after 1980.
  

To find the population in 1983, we plug 3 into the function where \(t = 3\), indicating three years after 1980, and compute:

• \( P(3) = 67.38 \cdot (1.026)^3 \)
• \( P(3) = 67.38 \cdot 1.080 \approx 72.74 \text{ million} \)

This representation not only helps in calculating numerical values efficiently but also aids in communicating mathematical processes clearly to others. Functional notation simplifies the manipulation and computation of complex mathematical expressions, ensuring consistency and accuracy when predicting values.

Calculating the growth rate is a crucial aspect of understanding exponential growth. It helps to quantify how quickly a population is increasing or decreasing.
For exponential growth, the rate is often expressed as a percentage that converts into a decimal for calculations. In this scenario, Mexico's population from 1980 to 1985 grew at a rate of 2.6% annually.
First, we transform the percentage growth rate into a decimal by dividing by 100, giving us a growth rate \(r\) of 0.026.

• The complete exponential growth function captures this rate as \( P(t) = 67.38 \cdot (1.026)^t \), where \(1.026 = 1 + 0.026\).

This part of the formula, \(1 + r\), signifies the annual compounded growth factor for the population. By raising this factor to the power of \(t\), we accurately model the cumulative effect of growth over multiple years.
Calculating how long it will take for the population to reach a specific size, like 90 million, further applies this growth rate calculation. By using algebraic manipulation and logarithms, we determine that the population reaches 90 million approximately in mid-1991, illustrating the power of growth rate in future projections.