Problem 66 - College Algebra and Calculus: An Applied Approach

Question

Population Growth The population \(P\) of a small city increases according to the model \(P(t)=36,000 e^{0.0156 t}\) where \(t\) represents the year, with \(t=0\) corresponding to 2000\. Use the model to predict the population in each year. (a) 2009 (b) 2011 (c) 2015 (d) 2018

Step-by-Step Solution

Answer

The predicted populations for the given years are approximated as follows:(a) 2009: 41,078(b) 2011: 42,923(c) 2015: 47,279(d) 2018: 51,006

Step 1: Calculate the population for 2009

Substitute \(t=9\) into the given model and calculate \(P(9)=36,000 e^{0.0156 * 9}\)

Step 2: Calculate the population for 2011

Like in step 1, substitute \(t=11\) into the given model and calculate \(P(11)=36,000 e^{0.0156 * 11}\)

Step 3: Calculate the population for 2015

Insert \(t=15\) into the given model to get the population for 2015 as \(P(15)=36,000 e^{0.0156 * 15}\)

Step 4: Calculate the population for 2018

Finally, substitute \(t=18\) into the model to find the population for 2018. This gives \(P(18)=36,000 e^{0.0156 * 18}\)

Key Concepts

Exponential Growth

Understanding exponential growth is crucial when studying populations. In essence, it describes a situation where the growth rate of a quantity is proportional to its current value. This means that the larger the population becomes, the faster it grows. Imagine a small snowball rolling down a hill, gathering more snow and gaining size and speed as it descends – this is a good analogy for exponential growth.

In the context of population growth, this concept helps in understanding how populations expand rapidly once they reach a certain size. Left unchecked or without limitations like space and resources, the population would continue to grow at an increasingly fast rate. The formula you encounter in population growth models, like the one for our small city, captures this behavior mathematically using an exponential function.

Mathematical Modeling

Mathematical modeling is a powerful tool that takes real-world scenarios and describes them through mathematical equations and formulas. In the example of population growth, the model provided (\( P(t) = 36,000 e^{0.0156 t} \)) uses an exponential function to predict the population over time.

The process of creating such a model often involves identifying trends, patterns, and relationships within the collected data. These models are not just educated guesses but are based on historical data and scientific principles that accurately represent the phenomenon being studied. It's important to remember that while models can be incredibly precise, they are simplifications and work within assumptions that may not always hold true in complex real-life situations.

Population Prediction

Population prediction involves using mathematical models to forecast future population sizes. The predictions are based on current and historical data, and the models take into account factors like birth rates, death rates, and migration patterns, although in a simpler model like the one we're looking at, these factors are all bundled into the growth rate.

Predictions are not certainties but provide an estimate of future trends. They are significant for planning purposes in urban development, resource management, and public services. By plugging the relevant year into our formula, future population sizes can be estimated, allowing for informed and proactive planning.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are commonly represented as \( f(x) = a \cdot b^{x} \) where \( a \) is the coefficient and \( b \) is the base, which in our population model is Euler's number \( e \) — a fundamental constant approximately equal to 2.71828. With exponential functions, the rate of change increases exponentially, which means it becomes faster as time goes on.

This is contrasted with linear growth, where the rate of change remains constant over time. Exponential functions are essential in modeling a variety of natural phenomena, including interest compounding in finance, radioactive decay in physics, and, of course, population growth in biology and demography.