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Chapter 5: Problem 12

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.

Short Answer

Expert verified
City A: \(P_{A}(t) = 1.2 \times (1.15)^{t/10}\), City B: \(P_{B}(t) = 1.2 \times (0.85)^{t/10}\).

Step by step solution

01

Understanding the problem

Two cities have populations of 1.2 million each. City A grows by a factor of 1.15 every 10 years while City B decays by a factor of 0.85 every 10 years. Need to find exponential functions for both cities and generate population tables and graphs over 50 years.
02

Formulating the exponential functions

The population of City A can be modeled as an exponential growth function: \[P_{A}(t) = 1.2 \times (1.15)^{t/10}\] The population of City B can be modeled as an exponential decay function: \[P_{B}(t) = 1.2 \times (0.85)^{t/10}\]
03

Generating the population table for City A

Calculate the population of City A at 10-year intervals: For \(t = 0\): \(P_{A}(0) = 1.2 \times (1.15)^{0} = 1.2\) million For \(t = 10\): \(P_{A}(10) = 1.2 \times (1.15)^{1} = 1.38\) million For \(t = 20\): \(P_{A}(20) = 1.2 \times (1.15)^{2} 鈮� 1.587\) million For \(t = 30\): \(P_{A}(30) = 1.2 \times (1.15)^{3} 鈮� 1.825\) million For \(t = 40\): \(P_{A}(40) = 1.2 \times (1.15)^{4} 鈮� 2.099\) million For \(t = 50\): \(P_{A}(50) = 1.2 \times (1.15)^{5} 鈮� 2.41\) million
04

Generating the population table for City B

Calculate the population of City B at 10-year intervals: For \(t = 0\): \(P_{B}(0) = 1.2 \times (0.85)^{0} = 1.2\) million For \(t = 10\): \(P_{B}(10) = 1.2 \times (0.85)^{1} = 1.02\) million For \(t = 20\): \(P_{B}(20) = 1.2 \times (0.85)^{2} 鈮� 0.867\) million For \(t = 30\): \(P_{B}(30) = 1.2 \times (0.85)^{3} 鈮� 0.737\) million For \(t = 40\): \(P_{B}(40) = 1.2 \times (0.85)^{4} 鈮� 0.627\) million For \(t = 50\): \(P_{B}(50) = 1.2 \times (0.85)^{5} 鈮� 0.533\) million
05

Plotting the graph

Plot the population values from the tables for both cities on the same graph. Use the x-axis for the time in years and the y-axis for the population in millions. Mark points for each time interval and draw smooth curves to represent the exponential growth for City A and exponential decay for City B.

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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 equal time intervals. In the given problem, City A's population grows by a factor of 1.15 every 10 years. This can be described by the exponential growth function \(P_{A}(t) = 1.2 \cdot (1.15)^{t/10}\). Here, \(1.2\) represents the initial population in millions, \(1.15\) is the growth factor, and \(t/10\) adjusts the time period to match the growth factor's interval. Exponential growth functions are characterized by a J-shaped curve on a graph, showing rapid increases over time.
exponential decay
Exponential decay happens when a quantity decreases by a fixed percentage over equal time intervals. For City B, the population decreases by a factor of 0.85 every 10 years. This is represented by the exponential decay function \(P_{B}(t) = 1.2 \cdot (0.85)^{t/10}\). Here, \(1.2\) is the initial population, \(0.85\) is the decay factor, and \(t/10\) adjusts the time for 10-year intervals. Graphically, exponential decay produces a downward-sloping curve, showing that the population reduces more slowly as time progresses.
population modeling
Population modeling uses mathematical functions to represent how a population changes over time. By using exponential functions for City A and City B, we can predict future populations based on current trends. This involves setting up the exponential functions based on growth or decay factors, as seen in the problem. For City A, the population model is \(P_{A}(t) = 1.2 \cdot (1.15)^{t/10}\), while for City B, it is \(P_{B}(t) = 1.2 \cdot (0.85)^{t/10}\). These models help us understand potential long-term impacts on resources, urban planning, and social services.
graphing exponential functions
Graphing exponential functions involves plotting values to visualize growth or decay over time. For City A and City B, we plotted population values at 10-year intervals from 0 to 50 years. The x-axis represents time in years, and the y-axis shows population in millions. For City A, we mark points such as \(P_{A}(0) = 1.2\) and \(P_{A}(10) 鈮� 1.38\). Connecting these points forms a curve that rises exponentially. For City B, points like \(P_{B}(0) = 1.2\) and \(P_{B}(10) 鈮� 1.02\) create a decreasing curve. These graphs visually demonstrate how populations change over time.
table of values
Creating a table of values helps organize data for graphing and understanding trends. For City A and City B, we list populations at specific time intervals. For instance, at \(t = 0\), both cities have populations of \(1.2\) million. At \(t = 50\), City A's population grows to around \(2.41\) million, while City B's dwindles to approximately \(0.533\) million. Tables like these make it easier to compare and visualize changes accurately. It helps to break down complex growth and decay into clear, understandable increments.

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Most popular questions from this chapter

The exponential function \(Q(T)=600(1.35)^{T}\) represents the growth of a species of fish in a lake, where \(T\) is measured in 5-year intervals. a. Determine \(Q(1), Q(2),\) and \(Q(3)\). b. Find another function \(q(t),\) where \(t\) is measured in years. c. Determine \(q(5), q(10),\) and \(q(15)\). d. Compare your answers in parts (a) and (c) and describe your results.

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.

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(Requires technology to find a best-fit function.) Reliable data on Internet use are hard to find, but World Telecommunications Indicators cites estimates of 3 million U.S. users in 1991,30 million in 1996,166 million in 2002,199 million in 2004 and 232 million in 2007 . a. Use technology to plot the data, and generate a best-fit linear and a best- fit exponential function for the data. Which do you think is the better model? b. What would the linear model predict for Internet usage in \(2010 ?\) What would the exponential model predict? c. Internet use: Go online and see if you can find the number of current internet users in the U.S. Which of your models turned out to be more accurate?
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