Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Classify the model as an exponential growth model or an exponential decay model.\(y=20 e^{-1.5 t}\)

Step-by-Step Solution

Answer

The given model \(y=20 e^{-1.5 t}\) is an exponential decay model.

Step 1: Identify the Format of the Model

The given model is in the form \(y=20 e^{-1.5 t}\). Comparing it to the general exponential model \(y=a e^{kt}\), the constants can be identified as: a = 20 (the initial amount) and k = -1.5 (the growth or decay rate).

Step 2: Classify the Model

Examine the growth/decay rate (-1.5). Since the rate is negative, the model corresponds to an exponential decay model. In an exponential decay model, the growth/decay rate is negative, which means that the quantity decreases over time.

Key Concepts

Exponential Growth

Exponential growth is a phenomenon where the quantity of something increases at a rate proportional to its current value. This pattern results in a rapid rise over time and is described by an exponential function where the growth rate is positive. For instance, when a population of bacteria doubles every hour, the number of bacteria grows exponentially because each hour the increase is based on the doubled amount from the previous hour, creating a steeper and steeper curve on a graph.

In terms of a mathematical model, exponential growth is typically represented by an equation in the form: \(y=ae^{kt}\), where \(a\) represents the initial amount, \(k\) is the positive growth rate constant, and \(t\) is the time that has passed. The key characteristic of exponential growth that distinguishes it from linear growth is that in exponential growth, the change is proportional to the current amount, not a fixed increment.

Exponential Functions

Exponential functions are mathematical expressions that describe changes where the rate of change is proportional to the value of the function at any point in time. They are defined by the equation \(y=ab^{t}\) or \(y=ae^{kt}\), where \(b\) and \(e\) (Euler's number, approximately 2.71828) are base constants, \(a\) is a constant representing the initial value, and \(k\) is the rate constant which can be positive for growth or negative for decay, a concept that will be explored further in the decay rate section.

One of the fascinating aspects of exponential functions is their 'J-shaped' growth curve when graphed. This curve starts off slowly, then accelerates rapidly as the function's value increases. These functions are crucial in various fields, from biology, describing population growth or decay, to finance, representing compound interest. Understanding the nature of exponential functions helps students recognize patterns of change that are not linear and anticipate the behavior of quantities that escalate in such a non-uniform manner.

Decay Rate

The decay rate in an exponential decay model is represented by a negative constant in the equation of an exponential function, indicating that the quantity is decreasing over time. In the context of the original exercise, the model \(y=20e^{-1.5t}\) illustrates exponential decay with its negative exponent on \(e\).

The decay rate, \(k\), affects the steepness of the decline in value; a more negative value of \(k\) means a faster decay. For the example given, with \(k = -1.5\), the 'half-life' or the time it takes for the quantity to reduce to half its initial value, can be calculated. The concept of half-life is commonly used in contexts like radioactive decay. A fundamental understanding of decay rate not only helps students decipher such real-world phenomena but also equips them with the ability to manipulate and apply exponential functions in various scientific and mathematical scenarios. It is the key to analyzing processes where diminishing patterns are present, such as cooling of substances, depreciation of assets, or even attenuation of sound.