Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Population Growth The population \(P\) of a town increases according to the model \(P(t)=4500 e^{0.0272 t}\) where \(t\) represents the year, with \(t=0\) corresponding to 2000 . Use the model to predict the population in each year. (a) 2010 (b) 2012 (c) 2015 (d) 2020

Step-by-Step Solution

Answer

After calculations, the predicted population is: (a) For 2010, the population is approximately 6051; (b) For 2012, it's approximately 6558; (c) For 2015, it's approximately 7690; (d) For 2020, it's approximately 9475.

Step 1: Understand and Use the Model

To solve this problem, apply the given exponential growth model \(P(t) = 4500e^{0.0272t}\). Since \(t=0\) in year 2000, to calculate the population of any year, use the difference between 2000 and the year in question for \(t\).

Step 2: Calculate for 2010

In this step, calculate for 2010. This means \(t = 2010 - 2000 = 10\). Substitute \(10\) for \(t\) in the equation: \(P(10) = 4500e^{0.0272*10}\). Calculate this equation to determine the population in 2010.

Step 3: Calculate for 2012

Now calculate for 2012. The value of \(t = 2012 - 2000 = 12\). Substitute this into the equation: \(P(12) = 4500e^{0.0272*12}\). Calculate this equation to find the population in 2012.

Step 4: Calculate for 2015

Next, for 2015 where \(t = 2015 - 2000 = 15\), substitute \(t\) in the equation: \(P(15) = 4500e^{0.0272*15}\). Solve this to determine the population in 2015.

Step 5: Calculate for 2020

Finally, for the year 2020 where \(t = 2020 - 2000 = 20\), replace \(t\) in the equation and solve: \(P(20) = 4500e^{0.0272*20}\). This will determine the population in 2020.

Key Concepts

Population Growth

Understanding the concept of population growth is essential in fields like biology, ecology, and urban planning. It refers to the increase in the number of individuals in a population over time. When studying population growth, one must consider factors like birth rates, death rates, and migration patterns, all of which influence the rate at which a population grows.

When describing population growth mathematically, models like the one presented in our exercise, are commonly used. In our example, the population of a town is modeled to grow exponentially, indicated by the term 'e' raised to a power that includes the time variable 't'. This model assumes that the growth rate is proportional to the current population, which tends to be accurate when resources are unlimited and the population size is not near carrying capacity.

However, it's important to recognize that this assumption may not hold true in real-world scenarios over long periods since constraints like food availability, space, and competition with other species can alter growth rates. Nonetheless, within certain timeframes and contexts, the exponential model offers a valuable approximation for predicting population growth.

Exponential Function

The exponential function is a mathematical expression where a constant base is raised to a variable exponent. In our textbook exercise, the base is the irrational number 'e', approximately equal to 2.71828, which is a fundamental constant in mathematics known for its unique properties when used in calculus and compounding formulas.

The general form of an exponential function is \( a \times b^{tx} \), where 'a' represents the initial amount, 'b' is the base of the exponential (in many natural processes, 'e' is used as the base due to natural growth patterns), and 'tx' is the exponent that typically incorporates the growth rate and time.

Exponential functions are widely used in various disciplines for modeling purposes. They capture scenarios where quantities increase rapidly and at a rate proportional to their current value, which describes many phenomena including population growth, radioactive decay, and compound interest in finance.

Mathematical Prediction

Mathematical prediction involves using models and equations to forecast future events or trends based on current or historical data. In the context of our population growth model, prediction involves applying the exponential function to make informed guesses about future population sizes. This practice is crucial for anticipating and preparing for changes, especially in contexts such as city planning, environmental conservation, and resource management.

By substituting different values of 't', representing different years, into our given model, one can predict the population size for those years. It's important to note, however, that predictions are only as good as the model and the assumptions upon which it is based. Therefore, while the exponential model can be a powerful predictive tool, it must be used with an understanding of its limitations and the potential for unforeseen variables to impact actual outcomes.