Problem 14 - Intermediate Algebra

Question

Determine whether each of the following functions model exponential growth or exponential decay. a. \(D(t)=150(0.44)^{t}\) b. \(H(t)=15,000(1.03)^{t}\)

Step-by-Step Solution

Answer

Function a models exponential decay, and function b models exponential growth.

Step 1: Understanding the Function Components

Start by identifying the base of the exponential function in each equation. Recall that in a function of the form \(f(t) = a \, b^t\), \(a\) is the initial value and \(b\) is the growth or decay factor.

Step 2: Analyze the Base for Function a

For the function \(D(t) = 150(0.44)^t\), the base is \(0.44\). Because \(0.44 < 1\), this function is an example of exponential decay.

Step 3: Analyze the Base for Function b

For the function \(H(t) = 15,000(1.03)^t\), the base is \(1.03\). Since \(1.03 > 1\), this function models exponential growth.

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases by a constant factor over equal time intervals. This happens when the base of the exponential function, often represented as \( b \) in the general form \( f(t) = a \cdot b^t \), is greater than 1. In the scenario of exponential growth, the larger the base, the faster the value of the function increases over time.

For the function described in the exercise, \( H(t) = 15,000(1.03)^t \), the base is \( 1.03 \), which is greater than 1, indicating exponential growth. Let's see why this signifies growth:

• The base of \( 1.03 \) means that with each passing unit of time, the quantity represented by \( H(t) \) grows by 3% over what it was in the previous time period.
• The initial value of \( 15,000 \) is where the growth starts.
• This kind of growth is common in scenarios like population growth, interest in bank accounts, or even viral content spreading online.

To summarize, whenever you have exponential growth, the base being greater than 1 ensures that the function value becomes larger and larger over time.

Exponential Decay

Exponential decay is the process where a quantity decreases by a consistent factor over identical time intervals. This takes place when the base \( b \) in the exponential form \( f(t) = a \cdot b^t \) is between 0 and 1. The smaller the base, the quicker the value diminishes over a given period.

In the example given, \( D(t) = 150(0.44)^t \), the base is \( 0.44 \), which is less than 1, signifying exponential decay:

• A base of \( 0.44 \) implies that with each unit of time, the quantity reduces to 44% of what it was in the previous time period.
• The initial value of \( 150 \) represents the starting point of this decay process.
• This kind of decrease is observed in real-world scenarios like radioactive decay, cooling of hot objects, or depreciation of assets.

Simply put, exponential decay ensures the exponential function's value becomes progressively smaller, eventually approaching zero but never quite reaching it.

Function Analysis

Analyzing an exponential function involves understanding its key components and what they indicate about the behavior of the function. The general formula \( f(t) = a \cdot b^t \) includes some critical elements:

• \(a\) is the initial value, representing the quantity at the starting time \( t = 0 \).
• \(b\) is the base, indicating whether the function demonstrates growth \((b > 1)\) or decay \((0 < b < 1)\).

Careful examination of these components tells you how a function behaves over time:

• In exponential growth, \(b\) determines the percentage increase each time period, making the function's value grow indefinitely as \( t \) increases.
• In exponential decay, \(b\) showcases the proportional decrease with time, leading the function's value to approach zero.

Function analysis doesn't just identify if an equation represents growth or decay; it also offers insights into the rate of change and potential real-world applications. Comprehending these nuances prepares you to predict outcomes and solve practical problems related to exponential functions seamlessly.