Problem 486 - Precalculus

Question

Determine whether the function\(y=156(0.825)^{t}\) represents exponential growth, exponential decay, or neither. Explain

Step-by-Step Solution

Answer

The function represents exponential decay because the base (0.825) is less than 1.

Step 1: Identify the General Form of Exponential Functions

Exponential functions are generally in the form \( y = a b^t \), where \( a \) is the initial amount, \( b \) is the base or growth/decay factor, and \( t \) is the time period. The function given, \( y = 156(0.825)^t \), matches this format.

Step 2: Determine the Value of the Base (b)

In the function \( y = 156(0.825)^t \), the base \( b \) is 0.825. This is the number raised to the power of \( t \).

Step 3: Check the Value of the Base

To determine if the function represents growth or decay, evaluate the base \( b \): - If \( b > 1 \), it's exponential growth.- If \( 0 < b < 1 \), it's exponential decay.- If \( b = 1 \), it represents neither growth nor decay.

Step 4: Analyze the Base (b = 0.825)

The base \( b = 0.825 \) is less than 1, indicating that the function represents exponential decay. This means as \( t \) increases, the value of \( y \) will decrease.

Key Concepts

Exponential Functions

An exponential function is a type of mathematical expression where a base number is raised to a variable exponent. These functions follow the general format:

• \( y = a \cdot b^t \)

In this equation, \( a \) stands for the initial value, \( b \) represents the base, and \( t \) is the time variable.
Exponential functions are notorious for their dramatic changes in value — they can rise or fall swiftly depending upon whether we are dealing with exponential growth or decay.
They are commonly seen in real-life scenarios, such as population growth, radioactive decay, and interest calculations. Each of these situations utilizes the power of exponential change in different ways.

Growth and Decay

Growth and decay differentiate two distinct behaviors within exponential functions:

• **Exponential Growth**: When the base \( b \) is greater than 1, it means that the function is growing exponentially. In other words, as time progresses, the output value of the function increases rapidly.
• **Exponential Decay**: When the base \( b \) is between 0 and 1, the function exhibits exponential decay. This suggests that over time, the value of the function decreases. This decay can occur in various contexts, such as reducing half-life in science.

In the given function \( y = 156(0.825)^t \), the base \( b = 0.825 \) is between 0 and 1, indicating exponential decay. Therefore, with time, the value of \( y \) diminishes, depicting a decay scenario.

Mathematical Modeling

Mathematical modeling involves using mathematical structures to represent real-life situations. Exponential functions are pivotal in these models because of their ability to depict rapid changes
For instance:

• Decay models can simulate depreciation in value, like the drop in price of a new car over time, or the fading of radioactivity in a sample.
• Growth models are used to illustrate things like the expansion of an investment with compound interest.

The process of modeling starts by identifying relevant variables and organizing them into an equation, such as the exponential model \( y = a \cdot b^t \). Determining the values of \( a \) and \( b \) helps in making accurate predictions about future behavior, capturing the essence of growth or decay in a quantifiable manner. By using such models, we gain valuable insights and can make informed decisions based on the data.