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Chapter 5: Problem 6

Match the statements (a) through (d) with the correct exponential function in (e) through (h). Assume time \(t\) is measured in the unit indicated. a. Radon-222 decays by \(50 \%\) every \(t\) days. b. Money in a savings account increases by \(2.5 \%\) per year. c. The population increases by \(25 \%\) per decade. d. The pollution in a stream decreases by \(25 \%\) every year. e. \(A=1000(1.025)^{t}\) f. \(A=1000(0.75)^{t}\) g. \(A=1000\left(\frac{1}{2}\right)^{t}\) h. \(A=1000(1.25)^{t}\)

Short Answer

Expert verified
a: g, b: e, c: h, d: f

Step by step solution

01

Identify the Type of Change (Decay or Growth)

Categorize each statement as either exponential decay or exponential growth. For example, statement (a) mentions a decrease, which indicates exponential decay. Statement (b) mentions an increase, indicating exponential growth.
02

Determine the Decay or Growth Factor

Convert the percentage changes into growth or decay factors. For decay: a decrease of 50% means the factor is 0.5. For growth: an increase of 2.5% means the factor is 1.025.
03

Match Statement (a) - Decay

Statement (a) mentions Radon-222 decays by 50% every t days, which corresponds to a decay factor of 0.5. Match with function (g): \[A=1000\left(\frac{1}{2}\right)^{t}\]
04

Match Statement (b) - Growth

Statement (b) mentions money increases by 2.5% per year, which corresponds to a growth factor of 1.025. Match with function (e): \[A=1000(1.025)^{t}\]
05

Match Statement (c) - Growth

Statement (c) mentions the population increases by 25% per decade, which corresponds to a growth factor of 1.25. Match with function (h): \[A=1000(1.25)^{t}\]
06

Match Statement (d) - Decay

Statement (d) mentions the pollution decreases by 25% every year, which corresponds to a decay factor of 0.75. Match with function (f): \[A=1000(0.75)^{t}\]

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Key Concepts

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

Exponential Growth
Exponential growth occurs when a quantity increases by a fixed percentage over a period of time. This type of growth can be observed in populations, investments, and other scenarios where the growth compounds over time. The general form of an exponential growth function can be written as:

\[A = A_0 (1 + r)^t\]

where:

  • A is the amount at time t
  • A_0 is the initial amount
  • 1 + r is the growth factor
  • r is the growth rate as a decimal
  • t is time
For instance, if you have a savings account that increases by 2.5% per year, the growth rate is 0.025. If the initial deposit is $1000, the compound interest over t years can be calculated as:

\[A = 1000(1.025)^t\].

This formula shows how the amount in the savings account increases exponentially over time.
Exponential Decay
Exponential decay describes the process where a quantity decreases by a fixed percentage over time. This type of behavior is often observed in radioactive substances, pharmaceuticals in the body, or pollutants in the environment. The general form of an exponential decay function can be expressed as:

\[A = A_0(1 - r)^t\]

where:

  • A is the amount remaining at time t
  • A_0 is the initial amount
  • 1 - r is the decay factor
  • r is the decay rate as a decimal
  • t is time
For example, Radon-222 decays by 50% every t days. Here, the decay rate is 0.50, and the decay factor is 1 - 0.50 = 0.5. If the initial amount of Radon-222 is 1000 units, the remaining amount after t days is:

\[A = 1000\left(\frac{1}{2}\right)^t\].

This formula helps to understand how the amount of substance decreases exponentially over time.
Growth Factor
The growth factor in exponential functions represents how much a quantity grows over a period. It is determined by adding 1 to the growth rate (expressed as a decimal). The formula for the growth factor is:

\[ \text{Growth Factor} = 1 + r \]

where:

  • \( r \) is the growth rate.
For instance, if a population increases by 25% per decade, the growth rate is 0.25. The growth factor then becomes:

\[ \text{Growth Factor} = 1 + 0.25 = 1.25 \]

Using this growth factor, the population's exponential growth can be modeled as:

\[A = A_0 (1.25)^t\],

where \(A_0\) is the initial population. This illustrates how the population grows exponentially over each decade.
Decay Factor
The decay factor in exponential functions indicates the fraction by which a quantity decreases over a time period. It is computed by subtracting the decay rate (in decimal form) from 1. The formula is:

\[ \text{Decay Factor} = 1 - r \]

where:

  • \( r \) stands for the decay rate.
For example, if pollution in a stream decreases by 25% every year, the decay rate is 0.25. The decay factor becomes:

\[ \text{Decay Factor} = 1 - 0.25 = 0.75 \]

When this decay factor is applied, the exponential decay can be modeled by:

\[A = A_0 (0.75)^t\],

where \(A_0\) is the initial amount of pollution. This formula helps illustrate how pollution decreases exponentially each year.

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Most popular questions from this chapter

In medicine and biological research, radioactive substances are often used for treatment and tests. In the laboratories of a large East Coast university and medical center, any waste containing radioactive material with a half-life under 65 days must be stored for 10 half-lives before it can be disposed of with the non-radioactive trash. a. By how much does this policy reduce the radioactivity of the waste? b. Fill out the accompanying chart and develop a general formula for the amount of radioactive pollution at any period, given an initial amount, \(A_{0}\).

[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?

Each of the following tables contains values representing either linear or exponential functions. Find the equation for each function. $$ \begin{aligned} &\text { a. }\\\ &\begin{array}{cccccc} \hline x & -2 & -1 & 0 & 1 & 2 \\ f(x) & 1.12 & 2.8 & 7 & 17.5 & 43.75 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { b. }\\\ &\begin{array}{cccccc} \hline x & -2 & -1 & 0 & 1 & 2 \\ g(x) & 0.1 & 0.3 & 0.5 & 0.7 & 0.9 \\ \hline \end{array} \end{aligned} $$

Mute swans were imported from Europe in the nineteenth century to grace ponds. Now there is concern that their population is growing too rapidly, edging out native species. Their population along the Atlantic coast has grown from 5800 in 1986 to 14,313 in 2002 . The increase is most acute in the mid-Atlantic region, but Massachusetts has also seen a jump, with 2939 mute swans counted in 2002 as compared with 585 in 1986 . a. Compare the growth factor in the mute swan population for the entire Atlantic coast with that for Massachusetts. b. Compare the average rate of change in the mute swan population for the entire Atlantic coast with that for Massachusetts. c. Construct both a linear and an exponential model for the mute swan population in Massachusetts since 1986 . d. Compare the projected populations of mute swans in Massachusetts by the year 2010 as predicted by your linear and exponential models.

Which of the following exponential functions represent growth and which decay? a. \(N=50 \cdot 2.5^{T}\) b. \(y=264(5 / 2)^{x}\) c. \(R=745(1.001)^{t}\) d. \(g(z)=\left(3 \cdot 10^{5}\right) \cdot(0.8)^{z}\) e. \(f(T)=\left(1.5 \cdot 10^{11}\right) \cdot(0.35)^{T}\) f. \(h(x)=2000\left(\frac{2}{3}\right)^{x}\)
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