Problem 17 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. \(y=11,701(0.97)^{t}\)

Step-by-Step Solution

Answer

The equation represents exponential decay.

Step 1: Identify the Base of the Exponential Expression

Examine the equation given: \(y = 11,701(0.97)^{t}\). The base of the exponential expression is \(0.97\).

Step 2: Determine the Nature of the Base

An exponential growth equation has a base greater than 1, while an exponential decay equation has a base between 0 and 1. In this equation, the base is \(0.97\), which is between 0 and 1.

Step 3: Conclude Based on the Nature of the Base

Since the base \(0.97\) is between 0 and 1, it indicates that the equation represents exponential decay.

Key Concepts

Exponential Growth

Exponential growth is when a quantity increases over time at a rate proportional to its current value. This kind of growth can be seen in populations, investments, and various natural processes. The hallmark of exponential growth is that the rate of change becomes faster as the quantity grows.

• The base of the exponential expression in exponential growth must be greater than 1.
• In mathematical terms, an exponential function is represented as \( y = a(b)^t \), where \( b > 1 \).
• The variable \( t \) usually represents time, showing how the quantity increases over each time period.

Consider a simple example: a population of bacteria that doubles in size every hour. If you start with 100 bacteria, after one hour, you'd have 200, after two hours, 400, and so on.
This continuous growth is what characterizes exponential growth, often described by the phrase "the rich get richer."

Exponential Decay

Exponential decay occurs when a quantity decreases at a rate proportional to its current value. Common examples include radioactive decay, cooling of an object, or depreciation of the value of an asset.

• The base of an exponential decay expression is between 0 and 1.
• Mathematically, the expression can be written as \( y = a(b)^t \), where \( 0 < b < 1 \).
• With each time period, the value of \( y \) becomes smaller, approaching zero but never quite reaching it.

Imagine you have a cup of hot coffee that cools down over time. Initially, it loses heat rapidly, but as it approaches room temperature, the rate of cooling slows down.
With our example equation \( y = 11,701(0.97)^{t} \), the base is 0.97, indicating that instead of growing, the quantity is shrinking. This pattern shows gradual decrease over time, signaling exponential decay.

Base of Exponential Expression

The base of an exponential expression plays a crucial role in defining the nature of the function. It's the number that is raised to a variable exponent, and it primarily determines whether the function will demonstrate growth or decay.

• If the base is greater than 1, it indicates exponential growth, where the quantity increases over time.
• If the base lies between 0 and 1, as seen in our test expression \( 0.97 \), it indicates exponential decay.

Determining the nature of the base is usually the first step in analyzing exponential functions. This analysis allows us to conclude the behavior of the function, whether it's heading progressively upward or downward.
To wrap up, the base sets the direction and the rate of change of the exponential function. It's key to understanding and predicting how things evolve within exponentially defined systems.