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Chapter 8: Problem 2

Is \(y=1.02^{t}\) an exponential decay model? Explain.

Short Answer

Expert verified
No, \(y=1.02^{t}\) is not an exponential decay model since the base of its exponent (1.02) is greater than 1.

Step by step solution

01

Recognizing Exponential Decay

An exponential decay model is generally in the form \(y = a(1-r)^{t}\) or \(y = ab^{-t}\) where \(a\) is the initial value, \(r\) is the rate of decay, \(b\) is a positive number greater than 1, and \(t\) is time. For both forms, the base of the exponent (which effectively is the factor by which the quantity diminishes over each time period) is less than 1.
02

Comparing the Model to the General Form

In the equation \(y=1.02^{t}\), comparing to the general forms described in step 1, the base of the exponent is 1.02, which is greater than 1.
03

Making a Conclusion

Therefore, since the base of the exponent in \(y=1.02^{t}\) is greater than 1, it does not meet the requirement for an exponential decay model (which would have a base of the exponent less than 1).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Decay
Exponential decay is a process where quantities decrease at a consistent percentage rate over equal time periods. It's commonly applied in real-world situations such as radioactive decay and cooling of substances. In mathematical terms, an exponential decay model is represented by the equation \( y = a(1 - r)^t \) or \( y = ab^{-t} \).

Here's what each symbol means:
  • \(a\) is the initial amount (also known as the starting value).
  • \(r\) is the decay rate (a positive number, but the equation uses \(1 - r\) to ensure the base is less than 1).
  • \(t\) stands for time, which could represent years, minutes, seconds, etc.
  • \(b\) in the alternative form is a number greater than 1, used with negative exponent \(-t\) to ensure the decay.
Key to understanding exponential decay is that the base, which is \(1 - r\), must be less than 1.
This ensures that the value of \(y\) decreases as \(t\) increases.
Exponential Growth
Exponential growth describes a situation where quantities increase in size at a consistent percentage rate over equal intervals of time. In everyday life, it can be seen in populations, finance with compound interest, and technologies.
The exponential growth model is typically expressed as \( y = a(1 + r)^t \).

Here's a breakdown:
  • \(a\) represents the initial value or starting amount.
  • \(r\) is the growth rate, a positive decimal (e.g., 3% growth rate is 0.03).
  • \(t\) refers to time.
The crucial point in exponential growth is the base, \(1 + r\), which must be greater than 1.
This ensures that the quantity \(y\) increases as time progresses.
In our example of \(y=1.02^{t}\), because 1.02 is greater than 1, it signifies exponential growth, not decay.
Mathematical Modeling
Mathematical modeling involves using mathematical structures, equations, and concepts to represent real-world phenomena. It is a valuable tool across various fields like science, engineering, and economics, helping to predict behavior and understand underlying dynamics.

To create a mathematical model:
  • Identify the system: Determine the key components, variables, and processes involved.
  • Choose a structure: Use appropriate mathematical tools like algebraic equations, calculus, or statistics that suit the nature of the system.
  • Validate the model: Compare the predictions and outputs of your model to real-world data to adjust and improve accuracy.
Exponential functions are commonly used in mathematical modeling when dealing with quantities that grow or decay proportionally to their size.
This includes scenarios like bank interest over time (growth) or radioactive decay (decay).
In modeling growth with the equation \(y=1.02^{t}\), we're representing how a quantity grows at a 2% rate per unit time.

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Most popular questions from this chapter

Use substitution to solve the system. $$\begin{aligned}&4 x-y=5\\\&2 x+4 y=15\end{aligned}$$

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Then evaluate the expression when \(a=1\) and \(b=2\). $$ \left(a^{3}\right)^{2} $$

Use a table of values to graph the equation. $$y=2$$
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