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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

The price of a home in Medford was \(\$ 100,000\) in 1985 and rose to \(\$ 200,000\) in 2005 . a. Create two models, \(f(t)\) assuming linear growth and \(g(t)\) assuming exponential growth, where \(t=\) number of years after \(1985 .\) b. Fill in the following table representing linear growth and exponential growth for \(t\) years after \(1985 .\) c. Which model do you think is more realistic?

A pollutant was dumped into a lake, and each year its amount in the lake is reduced by \(25 \%\). a. Construct a general formula to describe the amount of pollutant after \(n\) years if the original amount is \(A_{0}\). b. How long will it take before the pollution is reduced to below \(1 \%\) of its original level? Justify your answer.

MCI, a phone company that provides long-distance service, introduced a marketing strategy called "Friends and Family." Each person who signed up received a discounted calling rate to ten specified individuals. The catch was that the ten people also had to join the "Friends and Family" program. a. Assume that one individual agrees to join the "Friends and Family" program and that this individual recruits ten new members, who in turn each recruit ten new members, and so on. Write a function to describe the number of new people who have signed up for "Friends and Family" at the \(n\) th round of recruiting. b. Now write a function that would describe the total number of people (including the originator) signed up after \(n\) rounds of recruiting. c. How many "Friends and Family" members. stemming from this one person, will there be after five rounds of recruiting? After ten rounds? d. Write a 60 -second summary of the pros and cons of this recruiting strategy. Why will this strategy eventually collapse?

A bank compounds interest annually at \(4 \%\). a. Write an equation for the value \(V\) of \(\$ 100\) in \(t\) years. b. Write an equation for the value \(V\) of \(\$ 1000\) in \(t\) years. c. After 20 years will the total interest earned on \(\$ 1000\) be ten times the total interest earned on \(\$ 100 ?\) Why or why not?

The U.S. Department of Agriculture's data on per capita food commodity consumption for 1980 are listed in the following table. a. Using the data in the following table, construct exponential functions for each food category. Then evaluate each function for the year \(2000 .\) Assume \(t\) is the number of years since \(1980 .\) $$ \begin{array}{lccc} \hline & \begin{array}{c} \text { Per Capita } \\ \text { Consumption } \\ \text { (pounds) in } \\ 1980 \end{array} & \begin{array}{c} \text { Yearly } \\ \text { Growth/Decay } \\ \text { Factor } \end{array} & \begin{array}{c} \text { Exponential } \\ \text { Function } \end{array} \\ \hline \text { Beef } & 72.1 & 0.994 & B(t)= \\ \text { Chicken } & 32.7 & 1.024 & C(t)= \\ \text { Pork } & 52.1 & 0.996 & P(t)= \\ \text { Fish } & 12.4 & 1.010 & F(t)= \\ \hline \end{array} $$ b. Which commodities showed exponential growth? Which showed exponential decay? c. Write a 60 -second summary about the consumption of meat, chicken, and fish from 1980 to 2000 .
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