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Chapter 5: Problem 4

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.

Short Answer

Expert verified
(a) 1.002, (b) 0.937, (c) 1.23, (d) 4.0, (e) 0.65

Step by step solution

01

- Understanding Growth and Decay Factors

Growth or decay factors are calculated using the formula for exponential growth or decay: \[ \text{Factor} = 1 \text{ (+ or -) Rate} \text{ [expressed as a decimal]} \]For growth, use '+'. For decay, use '-'.
02

- Calculate Growth Factor for (a)

The weight increases by 0.2% every 5 days. Convert the percentage into a decimal:\[ \text{Rate} = 0.2 / 100 = 0.002 \]Then apply the formula for growth:\[ \text{Growth Factor} = 1 + 0.002 = 1.002 \]
03

- Calculate Decay Factor for (b)

The mass decreases by 6.3% every year. Convert the percentage into a decimal:\[ \text{Rate} = 6.3 / 100 = 0.063 \]Then apply the formula for decay:\[ \text{Decay Factor} = 1 - 0.063 = 0.937 \]
04

- Calculate Growth Factor for (c)

The population increases by 23% per decade. Convert the percentage into a decimal:\[ \text{Rate} = 23 / 100 = 0.23 \]Then apply the formula for growth:\[ \text{Growth Factor} = 1 + 0.23 = 1.23 \]
05

- Calculate Growth Factor for (d)

The profit increases by 300% per year. Convert the percentage into a decimal:\[ \text{Rate} = 300 / 100 = 3.0 \]Then apply the formula for growth:\[ \text{Growth Factor} = 1 + 3.0 = 4.0 \]
06

- Calculate Decay Factor for (e)

The blood alcohol level decreases by 35% per hour. Convert the percentage into a decimal:\[ \text{Rate} = 35 / 100 = 0.35 \]Then apply the formula for decay:\[ \text{Decay Factor} = 1 - 0.35 = 0.65 \]

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Key Concepts

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

growth factor
Growth factors are essential when dealing with exponential growth problems. They determine how much a quantity increases over a given period. The basic formula for finding a growth factor is: Growth Factor = 1 + Rate where the rate is expressed as a decimal. For instance, if a weight increases by 0.2% every 5 days, you first convert 0.2% to a decimal: Rate = 0.2 / 100 = 0.002 Then, apply the formula: Growth Factor = 1 + 0.002 = 1.002 This means the weight grows by a factor of 1.002 every 5 days. To make the conversion, always divide the percentage by 100 to turn it into a decimal.
decay factor
Decay factors are crucial when addressing exponential decay situations. They tell you how much a quantity decreases over time. The basic formula for finding a decay factor is: Decay Factor = 1 - Rate, where the rate is expressed as a decimal. For example, if a mass decreases by 6.3% every year, you convert 6.3% to a decimal: Rate = 6.3 / 100 = 0.063 Then, apply the formula: Decay Factor = 1 - 0.063 = 0.937 The mass decreases by a factor of 0.937 annually. Always remember to convert the percentage to a decimal by dividing by 100.
percentage conversion
Percentage conversion is the first step to solving growth and decay problems. This process involves turning a percentage into a decimal. To do this, simply divide the percentage by 100. For example: If you have a percentage of 23%, you'd convert it by calculating: 23 / 100 = 0.23 Similarly, for a percentage of 300%, you'd convert it by: 300 / 100 = 3.0 Once you have the decimal, you can use it in your growth or decay formulas. This conversion is a vital step for correct calculations.
exponential functions
Exponential functions describe growth or decay processes, where quantities change at a consistent rate. The general form of an exponential function is: y = a(1 + r)^t for growth or y = a(1 - r)^t for decay. where - y is the final amount - a is the initial amount - r is the growth (or decay) rate as a decimal - t represents time For example, if you start with a population of 1000 that grows by 23% per decade, the function is: y = 1000(1 + 0.23)^t Exponential functions help model real-world phenomena such as population growth, radioactive decay, and financial investments.

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Most popular questions from this chapter

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. \(y=21(3)^{-x}\)

Identify the doubling time or half-life of each of the following exponential functions. Assume \(t\) is in years. [Hint: What value of \(t\) would give you a growth (or decay) factor of 2 (or \(1 / 2\) )?] a. \(Q=70(2)^{t}\) b. \(Q=1000(2)^{t / 50}\) c. \(Q=300\left(\frac{1}{2}\right)^{t}\) d. \(Q=100\left(\frac{1}{2}\right)^{t / 250}\) e. \(N=550(2)^{t / 10}\) f. \(N=50\left(\frac{1}{2}\right)^{t / 20}\)

Which function has the steepest graph? $$ \begin{array}{l} F(x)=100(1.2)^{x} \\ G(x)=100(0.8)^{x} \\ H(x)=100(1.2)^{-x} \end{array} $$

In a chain letter one person writes a letter to a number of other people, \(N,\) who are each requested to send the letter to \(N\) other people, and so on. In a simple case with \(N=2\), let's assume person Al starts the process. Al sends to \(\mathrm{B} 1\) and \(\mathrm{B} 2 ; \mathrm{B} 1\) sends to \(\mathrm{C} 1\) and \(\mathrm{C} 2 ; \mathrm{B} 2\) sends to \(\mathrm{C} 3\) and \(\mathrm{C} 4\); and so on. A typical letter has listed in order the chain of senders who sent the letters. So \(\mathrm{D} 7\) receives a letter that has \(\mathrm{A} 1, \mathrm{~B} 2\), and \(\mathrm{C} 4\) listed. If these letters request money, they are illegal. A typical request looks like this: \(\cdot\) When you receive this letter, send \(\$ 10\) to the person on the top of the list. \(\cdot\) Copy this letter, but add your name to the bottom of the list and leave off the name at the top of the list. \(\cdot\) Send a copy to two friends within 3 days. For this problem, assume that all of the above conditions hold. a. Construct a mathematical model for the number of new people receiving letters at each level \(L,\) assuming \(N=2\) as shown in the above tree. b. If the chain is not broken, how much money should an individual receive? c. Suppose A 1 sent out letters with two additional phony names on the list (say Ala and Alb) with P.O. box addresses she owns. So both \(\mathrm{B} 1\) and \(\mathrm{B} 2\) would receive a letter with the list \(\mathrm{A} 1, \mathrm{~A} 1 \mathrm{a},\) Alb. If the chain isn't broken, how much money would Al receive? d. If the chain continued as described in part (a), how many new people would receive letters at level \(25 ?\) e. Internet search: Chain letters are an example of a "pyramid growth" scheme. A similar business strategy is multilevel marketing. This marketing method uses the customers to sell the product by giving them a financial incentive to promote the product to potential customers or potential salespeople for the product. (See Exercise \(31 .)\) Sometimes the distinction between multilevel marketing and chain letters gets blurred. Search the U.S. Postal Service website (www.usps.gov) for "pyramid schemes" to find information about what is legal and what is not. Report what you find.

[Part (e) requires use of the Internet and technology to find a best-fit function.] A "rule of thumb" used by car dealers is that the trade-in value of a car decreases by \(30 \%\) each year. a. Is this decline linear or exponential? b. Construct a function that would express the value of the car as a function of years owned. c. Suppose you purchase a car for \(\$ 15,000 .\) What would its value be after 2 years? d. Explain how many years it would take for the car in part (c) to be worth less than \(\$ 1000\). Explain how you arrived at your answer. e. Internet search: Go to the Internet site for the Kelley Blue Book (www.kbb.com). i. Enter the information about your current car or a car you would like to own. Specify the actual age and mileage of the car. What is the Blue Book value? ii. Keeping everything else the same, assume the car is I year older and increase the mileage by 10,000 . What is the new value? iii. Find a best-fit exponential function to model the value of your car as a function of years owned. What is the annual decay rate? iv. According to this function, what will the value of your car be 5 years from now?
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