/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 The populations of four towns fo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 4

The populations of four towns for time \(t,\) in years, are given by: $$ \begin{array}{l} P_{1}(t)=12,000(1.05)^{t} \\ P_{2}(t)=6000(1.07)^{t} \\ P_{3}(t)=100,000(1.01)^{t} \\ P_{4}(t)=1000(1.9)^{t} \end{array} $$ a. Which town has the largest initial population? b. Which town has the largest growth factor? c. At the end of 10 years, which town would have the largest population?

Short Answer

Expert verified
a. P3(t) has the largest initial population. b. P4(t) has the largest growth factor. c. P4(t) has the largest population at the end of 10 years.

Step by step solution

01

- Identify Initial Populations

Initial population is the population at time t = 0. For each function, the initial population is the constant coefficient.
02

- Compare Initial Populations

Compare the constant coefficients of the given functions: \(\begin{array}{l} P_{1}(t)=12,000(1.05)^{t} \ P_{2}(t)=6000(1.07)^{t} \ P_{3}(t)=100,000(1.01)^{t} \ P_{4}(t)=1000(1.9)^{t} \ End{array} \) Therefore the initial populations are 12000, 6000, 100000, and 1000 respectively. The largest initial population is from P3(t) = 100,000.
03

- Identify Growth Factors

Growth factor is the base of the exponential part of each function. For each function, this is the number inside the parentheses raised to the power of t.
04

- Compare Growth Factors

Compare the bases of the exponential expressions: (1.05), (1.07), (1.01), and (1.9). The largest growth factor is 1.9 from P4(t) = 1000(1.9)^{t}.
05

- Calculate Population after 10 Years

Substitute t = 10 into each function: \(鈥� P_{1}(10) = 12000(1.05)^{10}\) \(鈥� P_{2}(10) = 6000(1.07)^{10}\) \(鈥� P_{3}(10) = 100000(1.01)^{10}\) \(鈥� P_{4}(10) = 1000(1.9)^{10}\) Calculate the values: \(P_{1}(10) 鈮� 19561\) \(P_{2}(10) 鈮� 11836\) \(P_{3}(10) 鈮� 110462\) \(P_{4}(10) 鈮� 613106}\) Therefore, at the end of 10 years, P4(t) has the largest population.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

initial population
In population modeling, the initial population refers to the number of individuals in the population at the starting point, typically when time \( t \) is zero. This value provides a baseline for understanding growth over time.

In our exercise, each town's initial population is represented by the constant coefficient in their respective population functions:
  • \( P_1(t) = 12,000(1.05)^t \)
  • \( P_2(t) = 6000(1.07)^t \)
  • \( P_3(t) = 100,000(1.01)^t \)
  • \( P_4(t) = 1000(1.9)^t \)
Therefore, the initial populations are: 12,000 for Town 1, 6,000 for Town 2, 100,000 for Town 3, and 1,000 for Town 4. Comparing these, Town 3 has the largest initial population with 100,000 individuals.
growth factor
The growth factor in an exponential population model determines how the population changes over time. It is the value inside the parentheses in the population equation raised to the power of \( t \).

In our problem, the growth factors for each town are:
  • \( P_1(t) =12,000(1.05)^t \) - Growth Factor: 1.05
  • \( P_2(t) =6000(1.07)^t \) - Growth Factor: 1.07
  • \( P_3(t) =100,000(1.01)^t \) - Growth Factor: 1.01
  • \( P_4(t) =1000(1.9)^t \) - Growth Factor: 1.9
The town with the largest growth factor is clearly Town 4, with a growth factor of 1.9. This means that each year, the population in Town 4 grows by 90%.
exponential functions
Exponential functions are used to model situations where a quantity grows or decays at a rate proportional to its current value. The general form of an exponential function is:

\( P(t) = P_0 \cdot (b)^t \)

Where:
- \( P(t) \) is the population at time \( t \).
- \( P_0 \) is the initial population.
- \( b \) is the growth factor.
- \( t \) is the time.

In our given problem, the population of the towns is modeled using exponential functions of this form, showing how the population changes over time. The exponential nature indicates that the populations will grow faster as time increases, assuming the growth factor is greater than 1.
population modeling
Population modeling uses mathematical equations to represent how a population changes over time. Exponential models are a common choice for population growth, as they can show rapid increases or decreases based on the growth factor.

For example, the population models in our exercise are:
  • Town 1: \( P_1(t) = 12,000(1.05)^t \)
  • Town 2: \( P_2(t) = 6000(1.07)^t \)
  • Town 3: \( P_3(t) = 100,000(1.01)^t \)
  • Town 4: \( P_4(t) = 1000(1.9)^t \)
By substituting different values of \( t \), we can predict future populations. For instance, after 10 years:
\[ P_1(10) 鈮� 19,561 \]
\[ P_2(10) 鈮� 11,836 \]
\[ P_3(10) 鈮� 110,462 \]
\[ P_4(10) 鈮� 613,106 \]
This shows that Town 4 will have the largest population at the end of 10 years due to its high growth factor.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Each of two towns had a population of 12,000 in \(1990 .\) By 2000 the population of town A had increased by \(12 \%\) while the population of town B had decreased by \(12 \%\). Assume these growth and decay rates continued. a. Write two exponential population models \(A(T)\) and \(B(T)\) for towns A and \(\mathrm{B}\), respectively, where \(T\) is the number of decades since 1990 . b. Write two new exponential models \(a(t)\) and \(b(t)\) for towns A and \(\mathrm{B}\), where \(t\) is the number of years since 1990 . c. Now find \(A(2), B(2), a(20)\), and \(b(20)\) and explain what you have found.

A swimming pool is initially shocked with chlorine to bring the chlorine concentration to 3 ppm (parts per million). Chlorine dissipates in reaction to bacteria and sun at a rate of about \(15 \%\) per day. Above a chlorine concentration of 2 ppm, swimmers experience burning eyes, and below a concentration of 1 ppm, bacteria and algae start to proliferate in the pool environment. a. Construct an exponential decay function that describes the chlorine concentration (in parts per million) over time. b. Construct a table of values that corresponds to monitoring chlorine concentration for at least a 2-week period. c. How many days will it take for the chlorine to reach a level tolerable for swimmers? How many days before bacteria and algae will start to grow and you will need to add more chlorine? Justify your answers.

(Graphing program recommended.) Make a table of values and plot each pair of functions on the same coordinate system. a. \(y=2^{x}\) and \(y=2 x\) for \(-3 \leq x \leq 3\) b. \(y=(0.5)^{x}\) and \(y=0.5 x\) for \(-3 \leq x \leq 3\) c. Which of the four functions that you drew in parts (a) and (b) represent growth? d. How many times did the graphs that you drew for part (a) intersect? Find the coordinates of any points of intersection. e. How many times did the graphs that you drew for part (b) intersect? Find the coordinates of any points of intersection.

According to the Arkansas Democrat Gazette (February \(27,\) 1994): Jonathan Holdeen thought up a way to end taxes forever. It was disarmingly simple. He would merely set aside some money in trust for the government and leave it there for 500 or 1000 years. Just a penny, Holdeen calculated, could grow to trillions of dollars in that time. But the stash he had in mind would grow much bigger-to quadrillions or quintillions-so big that the government, one day, could pay for all its operations simply from the income. Then taxes could be abolished. And everyone would be better off. a. Holdeen died in 1967 , leaving a trust of \(\$ 2.8\) million that is being managed by his daughter, Janet Adams. In 1994 , the trust was worth \(\$ 21.6\) million. The trust was debated in Philadelphia Orphans' Court. Some lawyers who were trying to break the trust said that it is dangerous to let it go on, because "it would sponge up all the money in the world." Is this possible? b. After 500 years, how much would the trust be worth? Would this be enough to pay off the current national debt (over \(\$ 7\) trillion in 2004\() ?\) What about after 1000 years? Describe the model you used to make your predictions.

Between 1970 and 2000 , the United States grew from about 200 million to 280 million, an increase of approximately \(40 \%\) in this 30 -year period. If the population continues to expand by \(40 \%\) every 30 years, what will the U.S. population be in the year \(2030 ?\) In \(2060 ?\) Use technology to estimate when the population of the United States will reach I billion.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.