Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 1
Identify and interpret the decay factor for each of the following functions: a. \(P=450(0.43)^{t}\) b. \(f(t)=3500(0.95)^{t}\) c. \(y=21(3)^{-x}\)
Short Answer
Expert verified
a. 0.43, b. 0.95, c. \(\frac{1}{3}\) or 0.333
Step by step solution
01
- Identify Decay Factor for Function a
The function given is \(P=450(0.43)^t\). Here, the base of the exponent, 0.43, is the decay factor.
02
- Interpret the Decay Factor for Function a
The decay factor, 0.43, means that the quantity decreases to 0.43 times its previous amount for each unit increase in time, t.
03
- Identify Decay Factor for Function b
The function given is \(f(t)=3500(0.95)^t\). Here, the base of the exponent, 0.95, is the decay factor.
04
- Interpret the Decay Factor for Function b
The decay factor, 0.95, means that the quantity decreases to 0.95 times its previous amount for each unit increase in time, t.
05
- Identify Decay Factor for Function c
The function given is \(y=21(3)^{-x}\). First, rewrite the function to \(y=21\left(\frac{1}{3}\right)^{x}\). Now, the base of the exponent, \(\frac{1}{3}\) or approximately 0.333, is the decay factor.
06
- Interpret the Decay Factor for Function c
The decay factor, \(\frac{1}{3}\) or approximately 0.333, means that the quantity decreases to \(\frac{1}{3}\) of its previous amount for each unit increase in time, x.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Decay
Exponential decay happens when a quantity decreases by a consistent percentage over equal intervals of time. In contrast to exponential growth, where the quantity increases over time, exponential decay reduces the quantity gradually. This is seen in various real-world phenomena such as radioactive decay, cooling of objects, and depreciation of assets. Each function representing exponential decay includes a base less than 1, which reduces the initial value step-by-step as time progresses.
For example, consider the function from the exercise, \(P=450(0.43)^t\). The quantity 450 decreases to 43% of its value every unit of time.
For example, consider the function from the exercise, \(P=450(0.43)^t\). The quantity 450 decreases to 43% of its value every unit of time.
Decay Factor Interpretation
The decay factor is crucial for understanding how quickly a quantity decreases over time. It is the base of the exponent in an exponential function. When the base is less than 1, we have a decay factor, which indicates the rate of decay. For the function \(P=450(0.43)^t\), the decay factor is 0.43.
This means that for each time unit that passes, the quantity P is multiplied by 0.43. If you start with 450, after one unit of time, the quantity reduces to 0.43 * 450, after two units of time, it reduces to 0.43 * (0.43 * 450), and so on. Understanding the decay factor helps in making predictions about future values and understanding the nature of the decay.
This means that for each time unit that passes, the quantity P is multiplied by 0.43. If you start with 450, after one unit of time, the quantity reduces to 0.43 * 450, after two units of time, it reduces to 0.43 * (0.43 * 450), and so on. Understanding the decay factor helps in making predictions about future values and understanding the nature of the decay.
Mathematical Functions
Mathematical functions, specifically exponential functions, describe relationships where quantities grow or decay at rates proportional to their current value. In algebra, these are usually written in the form \(P = P_0e^{kt}\) or \(y = ab^x\). For exponential decay, we focus on \(y = ab^x\) with \(0 < b < 1\).
The decay factor b determines the rate of decrease. If b is close to 1, the decay is slow; if b is much less than 1, the decay is rapid. Recognizing these characteristics allows students to interpret and employ exponential functions effectively across various scenarios, including scientific phenomena and financial depreciation.
The decay factor b determines the rate of decrease. If b is close to 1, the decay is slow; if b is much less than 1, the decay is rapid. Recognizing these characteristics allows students to interpret and employ exponential functions effectively across various scenarios, including scientific phenomena and financial depreciation.
Exponential Growth and Decay Analysis
To analyze exponential growth and decay, it鈥檚 essential to identify the base of the exponent and determine whether it's greater than or less than 1. For growth, the base (or growth factor) exceeds 1, leading to an increasing quantity. For decay, as studied in the problem, the base (or decay factor) is less than 1, leading to a decreasing quantity.
For example, in the function \(f(t)=3500(0.95)^t\), the base of 0.95 signifies decay. This allows us to project the value of f(t) for future time points by continually applying the decay factor. Understanding these principles aids in making sense of data trends and forecasting future behaviors based on current models.
For example, in the function \(f(t)=3500(0.95)^t\), the base of 0.95 signifies decay. This allows us to project the value of f(t) for future time points by continually applying the decay factor. Understanding these principles aids in making sense of data trends and forecasting future behaviors based on current models.
Algebra Education
In algebra education, learning about exponential decay is foundational for students. It introduces them to exponential functions and their applications. By working through exercises like the given problem, students develop intuition for how quantities change over time in real-world contexts.
It also builds their ability to manipulate and interpret functions, enhancing problem-solving skills. Teaching these concepts using clear, real-life examples, as shown in the solutions, and practicing different scenarios, positions students to advance to more complex subjects in calculus and beyond. This comprehensive understanding is pivotal for academic success in mathematics and sciences.
It also builds their ability to manipulate and interpret functions, enhancing problem-solving skills. Teaching these concepts using clear, real-life examples, as shown in the solutions, and practicing different scenarios, positions students to advance to more complex subjects in calculus and beyond. This comprehensive understanding is pivotal for academic success in mathematics and sciences.
