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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.

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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.
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.
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.
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.
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.

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Most popular questions from this chapter

The following exponential functions represent population growth. Identify the initial population and the growth factor. a. \(Q=275 \cdot 3^{T}\) b. \(P=15,000 \cdot 1.04^{t}\) c. \(y=\left(6 \cdot 10^{8}\right) \cdot 5^{x}\) d. \(A=25(1.18)^{t}\) e. \(P(t)=8000(2.718)^{t}\) f. \(f(x)=4 \cdot 10^{5}(2.5)^{x}\)

What is the growth or decay factor for each given time period? a. Weight increases by \(0.2 \%\) every 5 days. b. Mass decreases by \(6.3 \%\) every year. c. Population increases \(23 \%\) per decade. d. Profit increases \(300 \%\) per year. e. Blood alcohol level decreases \(35 \%\) per hour.

(Graphing program recommended.) Cosmic ray bombardment of the atmosphere produces neutrons, which in turn react with nitrogen to produce radioactive carbon-14. Radioactive carbon-14 enters all living tissue through carbon dioxide (via plants). As long as a plant or animal is alive, carbon-14 is maintained in the organism at a constant level. Once the organism dies, however, carbon-14 decays exponentially into carbon-12. By comparing the amount of carbon- 14 to the amount of carbon-12, one can determine approximately how long ago the organism died. Willard Libby won a Nobel Prize for developing this technique for use in dating archaeological specimens. The half-life of carbon-14 is about 5730 years. In answering the following questions, assume that the initial quantity of carbon- 14 is 500 milligrams. a. Construct an exponential function that describes the relationship between \(A,\) the amount of carbon- 14 in milligrams, and \(t,\) the number of 5730 -year time periods. b. Generate a table of values and plot the function. Choose a reasonable set of values for the domain. Remember that the objects we are dating may be up to 50,000 years old. c. From your graph or table, estimate how many milligrams are left after 15,000 years and after 45,000 years. d. Now construct an exponential function that describes the relationship between \(A\) and \(T,\) where \(T\) is measured in years. What is the annual decay factor? The annual decay rate? e. Use your function in part (d) to calculate the number of milligrams that would be left after 15,000 years and after 45,000 years.

Each table has values representing either linear or exponential functions. Find the equation for each function. $$ \begin{array}{cccccc} \hline x & -2 & -1 & 0 & 1 & 2 \\ h(x) & 160 & 180 & 200 & 220 & 240 \\ \hline \end{array} $$ $$ \begin{aligned} &\text { b. }\\\ &\begin{array}{cccccc} \hline x & 0 & 10 & 20 & 30 & 40 \\ j(x) & 200 & 230 & 264.5 & 304.17 & 349.8 \\ \hline \end{array} \end{aligned} $$

(Graphing program recommended.) On the same graph, sketch \(f(x)=3(1.5)^{x}, g(x)=-3(1.5)^{x},\) and \(h(x)=3(1.5)^{-x}\) a. Which graphs are mirror images of each other across the \(y\) -axis? b. Which graphs are mirror images of each other across the \(x\) -axis? c. Which graphs are mirror images of each other about the origin (i.e., you could translate one into the other by reflecting first about the \(y\) -axis, then about the \(x\) -axis)? d. What can you conclude about the graphs of the two functions \(f(x)=C a^{x}\) and \(g(x)=-C a^{x} ?\) e. What can you conclude about the graphs of the two functions \(f(x)=C a^{x}\) and \(g(x)=C a^{-x} ?\)
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