Problem 18 - Algebra 1

Question

Classify the model as exponential growth or exponential decay. Identify the growth or decay factor and the percent of increase or decrease per time period. $$y=14(0.98)^{t}$$

Step-by-Step Solution

Answer

The model given, \(y=14(0.98)^{t}\), represents exponential decay. The decay factor is \(0.98\), which means that there is a \(2\% \) decrease each time period.

Step 1: Identify the type of model

The given model is \(y=14(0.98)^{t}\). As \(0.98\) is less than \(1\), and the exponent is \(t\), the model can be identified as exponential decay.

Step 2: Determine the decay factor

The decay factor is the base of the exponent, which is \(0.98\). The fact that this value is less than \(1\) represents the quantity reduced in each time period.

Step 3: Calculate the Percentage Decrease

The decay factor is a subtraction from \(1\), and the difference can be represented as a percentage decrease. In this case, the difference is \(1-0.98=0.02\) or \(2\% \). Thus, there is a \(2\% \) decrease per time period.

Key Concepts

Decay Factor

Understanding the concept of a decay factor provides the foundation for exponential decay models. In the context of our exercise, the decay factor is represented by the base of the exponent in the equation \(y=14(0.98)^{t}\). This value, which is 0.98, is crucial as it determines the rate at which the quantity is decreasing over each time period.

To elaborate, an exponential decay model takes the form \(y=ae^{(-kt)}\) or \(y=ab^{t}\), where \(a\) is the initial amount, \(b\) is the decay factor (with \(0

• The decay factor tells us by what proportion the original amount is multiplied at each step or time period.
• In our case, for every time unit that passes, the quantity represented by \(y\) will shrink to 98% of its size in the previous time unit, due to the decay factor of 0.98.

As important as it is to identify this factor, it is also crucial to understand and explain that as the decay factor is less than one, our model inherently depicts a process of exponential decay, not growth.

Percent Decrease

The percent decrease in an exponential decay model is a measure of how much a quantity is reduced per unit of time. To find the percent decrease, we initially need to substract the decay factor from 1 and then express the result as a percentage.

In our example, the calculation is quite simple: The decay factor is 0.98, so the decrease per time period is \(1 - 0.98 = 0.02\), which we can convert to a percentage by multiplying by 100. That gives us a 2% decrease per time period. This percentage is a key indicator of the rate at which the quantity represented by the model is fading away over time.

• This representation allows students to conceptualize the decrease in more tangible terms, providing a clearer picture of what exponential decay means in practical situations.
• Understanding this concept is crucial for real-world applications, such as calculating the depreciation of assets, the dispersion of heat, or the decay of radioactive substances.
• When providing explanations, it's beneficial to include examples from everyday life to help students relate the theoretical concept to their experiences and observations.

A detailed and real-world analogy would be to compare the percent decrease to a discount in a store - it tells you how much less you have to pay compared to the original price, or in our mathematical context, how much smaller the quantity becomes.

Exponential Growth vs Decay

Distinguishing between exponential growth and decay is essential for correctly interpreting mathematical models, and it is characterized by the value of the base in the exponential function. With exponential growth, the quantity increases over time, while in exponential decay, it decreases.

• In equations representing exponential growth, such as \(y=ab^{t}\), the base \(b\) will be greater than 1.
• Conversely, for decay, like the one presented in our exercise \(y=14(0.98)^{t}\), the base is less than 1.

One way we can help students visualize this is by comparing it to populations. If a population of bacteria is growing exponentially, it multiplies at a consistent rate, leading to a rapid increase in the number of bacteria. Think of it like compound interest in a bank account growing over time.

On the flip side, if the bacterial population was shrinking due to an antibiotic, we'd expect to see the number of bacteria decline at a steady rate, similar to our principle of exponential decay in the exercise. This analogy can be linked back to the concept of percent decrease to further illustrate the rate at which the population decreases.
It's vital to communicate these differences clearly, using real-life examples that resonate with students, and ensuring they understand the implications of each type of model for predicting and analyzing real-world phenomena.