Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Classify the model as an exponential growth model or an exponential decay model.\(y=4 e^{0.07 t}\)

Step-by-Step Solution

Answer

The model \(y=4 e^{0.07 t}\) is an exponential growth model.

Step 1: Identify Model

For an equation of the form \(y= a.e^{kt}\), if the base \(e > 1\) and exponent \(kt > 0\) we have an exponential growth model. If the exponent is less than zero, \(kt < 0\), we have an exponential decay model. In the given equation \( y = 4.e^{0.07.t}\), the base is \(e\), and exponent is \(0.07t\) .

Step 2: Determine Model

Since the base of the exponent (which is \(e\)) is greater than 1 and \(0.07t > 0\) for any positive value of \(t\), we can conclude that this model represents exponential growth.

Key Concepts

Exponential Functions

Exponential functions represent a specific form of mathematical equation where a constant base is raised to a variable exponent. The general form of an exponential function is \( y = a \times b^{x} \), where \( a \) is the initial value, \( b \) is the base and must be a positive real number not equal to 1, and \( x \) is the exponent. The base \( e \) (approximately 2.718) is a mathematical constant and is commonly used as the base for natural exponential functions, \( y = a.e^{kt} \).

Exponential functions are characterized by their rate of growth or decay, which is determined by the exponent. When the exponent increases over time, we witness an exponential growth; conversely, a decreasing exponent signifies exponential decay. Exponential functions are ubiquitous across different fields, including biology for population growth, finance for compounding interest, and physics for radioactive decay. Understanding the direction and rate of change in such functions is crucial for interpreting and predicting behaviors in various scenarios.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario to analyze, explain, or predict its behavior. It serves as a potent tool in understanding complex systems and making informed decisions. Models can range from simple linear relationships to intricate differential equations.

In the context of exponential functions, mathematical models allow us to project the behavior of phenomena that change at rates proportional to their own size. For the model in question, \( y=4e^{0.07t} \), it's derived from the exponential function format for continuous growth or decay. The constant \( 4 \) signifies the starting quantity, while the exponent includes a rate \( 0.07 \) and a variable \( t \) that usually stands for time. By manipulating the variables and constants in these models, we can simulate different scenarios and outcomes, making them a valuable asset in multiple disciplines such as economics, ecology, and engineering.

Exponential Growth

Exponential growth refers to the increase in quantity at a rate proportional to its current value, meaning that as the quantity grows, it does so at a faster and faster rate. This is characteristic of processes where the amount of change is directly linked to the current state. In the context of the given equation \( y = 4e^{0.07t} \), the exponent represents the growth rate, and since it is positive, this function exemplifies exponential growth.

In real-world scenarios, populations of organisms, investments, and even viral social media posts often exhibit exponential growth. For example, if a population of bacteria doubles every hour, the number of bacteria is an exponential function of time. Recognizing exponential growth patterns helps in anticipating how quickly a system may expand and is critical for planning and resource management, especially in contexts like environmental science and business.