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Chapter 5: Problem 3

Given the following exponential decay functions, identify the decay rate in percentage form. a. \(Q=400(0.95)^{t}\) b. \(A=600(0.82)^{\mathrm{r}}\) c. \(P=70,000(0.45)^{t}\) d. \(y=200(0.655)^{x}\) e. \(A=10(0.996)^{T}\) f. \(N=82(0.725)^{T}\)

Short Answer

Expert verified
(a) 5%, (b) 18%, (c) 55%, (d) 34.5%, (e) 0.4%, (f) 27.5%

Step by step solution

01

Identify decay rate formula

The decay rate can be obtained from the base of the exponential function. If the function is of the form \(Q = P( b )^{ t } \) where \(b\) is the base, the decay rate \( r \) is \( r = 1 - b \). The decay rate (in percentage) is found by multiplying \(r\) by 100.
02

Calculate decay rate for function (a)

Given the function: \(Q = 400(0.95)^{t} \). The base \(b\) is 0.95. The decay rate \(r\) is calculated as \( r = 1 - 0.95 = 0.05 \). Convert to percentage: \(0.05 \times 100 = 5\text{\textpercent} \).
03

Calculate decay rate for function (b)

Given the function: \(A = 600(0.82)^{\text{r}} \). The base \(b\) is 0.82. The decay rate \(r\) is calculated as \( r = 1 - 0.82 = 0.18 \). Convert to percentage: \(0.18 \times 100 = 18\text{\textpercent} \).
04

Calculate decay rate for function (c)

Given the function: \( P = 70,000(0.45)^{ t } \). The base \(b\) is 0.45. The decay rate \(r\) is calculated as \( r = 1 - 0.45 = 0.55 \). Convert to percentage: \(0.55 \times 100 = 55\text{\textpercent} \).
05

Calculate decay rate for function (d)

Given the function: \( y = 200(0.655)^{ x } \). The base \(b\) is 0.655. The decay rate \(r\) is calculated as \( r = 1 - 0.655 = 0.345 \). Convert to percentage: \(0.345 \times 100 = 34.5\text{\textpercent} \).
06

Calculate decay rate for function (e)

Given the function: \( A = 10(0.996)^{ T } \). The base \(b\) is 0.996. The decay rate \(r\) is calculated as \( r = 1 - 0.996 = 0.004 \). Convert to percentage: \(0.004 \times 100 = 0.4\text{\textpercent} \).
07

Calculate decay rate for function (f)

Given the function: \( N = 82(0.725)^{ T } \). The base \(b\) is 0.725. The decay rate \(r\) is calculated as \( r = 1 - 0.725 = 0.275 \). Convert to percentage: \(0.275 \times 100 = 27.5\text{\textpercent} \).

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Key Concepts

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

Decay Rate
In exponential decay, the **decay rate** refers to how rapidly a quantity decreases over time. The general format of an exponential decay function is \(Q = P( b )^{ t }\), where **b** is the base. To find the decay rate **r**, you subtract the base from 1: \( r = 1 - b \). This gives the rate in its decimal form. To express it as a percentage, multiply by 100. For example, if **b** is 0.95, the rate **r** is \(1 - 0.95 = 0.05\), which converts to 5%.
Exponential Functions
An **exponential function** describes a situation where a quantity changes at a rate proportional to its current value. This creates a curve that either grows or decays exponentially. When dealing with decay, the function decreases over time and takes the form \(Q = P( b )^{ t }\), where:
  • **Q** is the final amount.
  • **P** is the initial amount.
  • **b** is the decay factor (always between 0 and 1).
  • **t** is the time variable.
Understanding how to manipulate and interpret these functions is crucial in fields like biology, physics, and finance.
Percentage Conversion
To express a decay rate as a **percentage**, you simply convert its decimal form by multiplying by 100. This makes it easier to understand and communicate. For instance, if a decay rate is 0.18, converting this to a percentage involves:
\(0.18 \times 100 = 18\text{\%}\)
It's a straightforward yet powerful conversion that makes numerical data more accessible. Be careful to always use the correct order of operations and ensure that you're converting from the true decimal form.
Algebraic Formulas
Understanding and manipulating **algebraic formulas** is essential for solving exponential decay problems. These formulas allow us to isolate variables and calculate unknown quantities. For instance, given \(Q = P( b )^{ t }\), to find decay rate, we use
\( r = 1 - b \)
and then convert it accordingly. Mastery over these manipulations helps in various applications like exact timing in financial depreciations, radioactive decay in physics, and understanding biological processes over time. Always ensure to follow the appropriate algebraic rules and steps.

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Most popular questions from this chapter

Lead- 206 is not radioactive, so it does not spontaneously decay into lighter elements. Radioactive elements heavier than lead undergo a series of decays, each time changing from a heavier element into a lighter or more stable one. Eventually, the element decays into lead- 206 and the process stops. So, over billions of years, the amount of lead in the universe has increased because of the decay of numerous radioactive elements produced by supernova explosions. Radioactive uranium- 238 decays sequentially into thirteen other lighter elements until it stabilizes at lead-206. The half-lives of the fifteen different elements in this decay chain vary from 0.000164 seconds (from polonium- 214 to lead- 210 ) all the way up to 4.47 billion years (from uranium- 238 to thorium- 234 ). a. Find the decay rate per billion years for uranium- 238 to decay into thorium- 234 . b. Find the decay rate per second for polonium-214 to decay into lead-2.10.

[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?

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.

Two cities each have a population of 1.2 million people. City A is growing by a factor of 1.15 every 10 years, while city \(\mathbf{B}\) is decaying by a factor of 0.85 every 10 years. a. Write an exponential function for each city's population \(P_{A}(t)\) and \(P_{B}(t)\) after \(t\) years. b. For each city's population function generate a table of values for \(x=0\) to \(x=50,\) using 10 -year intervals, then sketch a graph of each town's population on the same grid.

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.
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