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EARTH STUDIES

Introductory Module

World Population

Introduction

In this module, you will be introduced to the format and methods that will be used in the analysis of data and solution of problems that are presented in Earth Studies.  The Studies will all be presented in a somewhat structured format that will be outlined in this module.  Applets, or little programs, have been designed to be used with each Study; these will perform the mathematical operations you need to analyze data and build mathematical modules.  Also, links are included throughout each study that will take you to explanations of mathematical terms and methods.  Here you will be guided through a study on population that will illustrate the most common methods used in all of the Earth Math Studies and will also illustrate the use of the applets.  First, we outline the general structure for all the Studies.

Each Study will have three components:

  • the text, which will provide information about a particular issue and ask questions;
  • the Menu will be in two parts: the Tool Chest, which will contain the applets to perform the mathematical operations; and a Topics list, which will contain links to explanations of relevant mathematical topics;
  • the journal, which will be the place where you can record your thoughts, answers, and mathematical solutions.

The text will be presented in four sections: Comprehension; Acquisition; Application; and Reflection

  • The Comprehension section will provide information about the issue and ask pertinent questions for you to think about and express your own ideas regarding the issue.
  • The Acquisition section will provide more detailed information, relevant data, or direct you to a website to acquire more information or data.
  • The Application section will ask specific questions which will require mathematical analysis that will lead to a mathematical model.  The model should be able to provide answers to questions raised regarding the issue.  It is in this section that the applets will be used.
  • The Reflection section will ask questions regarding the reasonableness or validity of your model.

The Tool Chest will be the place where you can access the applets that will perform the mathematics.  Here you can enter your data or other quantitative information and use the applets to analyze data and construct mathematical models.  The results should answer the questions involved in the issue.  

The journal will be the place you record your answers to questions from the text and include the explanations of the mathematics you used.  Your journal should be written as a file in the word processing package on your computer; this will be the document you turn in to your instructor.

EARTH STUDY: WORLD POPULATION

Comprehension

In this section, questions are posed that are designed to stimulate your thoughts and observations regarding population increase/decrease, reasons for studying population change, and methods that might be used.  At this point, you should open a blank document in your word processor and record your answers to these questions.  Copy each question into the journal and type your response below in a different color or font.  Be sure to write in complete sentences and express your ideas so that others can understand.  Save your work and keep this window open throughout this study so you can easily record other questions and answers.  

Questions

A. Do you think that the population of the World is increasing or decreasing?

B. What have you noticed recently that led you to your answer to A?

C.  How do you think population change in the world might influence your life?

D. How do you think mathematics can be used to study population change?

E. What are some reasons for studying population change?

Acquisition

In this section, you will be given information about population increase, data is provided, and links to websites that contain more information and updated data. 

Estimates of world population before the twentieth century vary widely, but most sources put the number of people in the world in 1750 at about 750 million.  By mid-nineteenth century the population reached 1 billion and until 1930 the growth was never more than 1%; however, since 1950 the increase has never fallen below 1.6%.  Current patterns of population growth and the accompanying changes in consumption are placing increasing stresses on ecosystems through environmental degradation, deforestation, loss of biological diversity, over harvesting, and accumulation of toxic wastes. 

The Table below gives the United Nations estimates of the world population (in billions) every five years from 1950 -1995. 

World Population

Year
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995

Population

(billions)

2.554
2.779
3.038
3.345
3.704
4.086
4.457
4.856
5.295
5.732

World Watch 1996

In this section, objectives and assumptions are stated; these will determine your mathematical model for population.  You will use relevant mathematical tools that are provided on the applets to perform the analysis on the data and build your models.  You will develop a model for world measure population growth.  Applets can be accessed by clicking  “Tool Chest” on the Menu to the left of the screen.

You should follow the steps outlined after each question to familiarize yourself with the techniques used and the applets available.  Record answers in your journal.

The objectives and assumptions are listed below.

Objectives     

To determine:

1.  a linear model for world population;

2. the annual rate of change of world population.

Assumption

The current trends for world population continue.

Application

Linear Models    

In this part, you will find a linear function to approximate data for world population and use this to

•     estimate annual population growth,

•     estimate the population for years other than those in the data set, and

•     forecast future population size.

(Round off to three places for this work.)

Problem Set

The applets needed needed to complete this problem are provided in the text for this Introduction module. Usually, you will need to follow these steps.

  • Scroll down to “Tool Chest” on the menu to the left of the screen. 
  • Click “linear regression” (since we are looking for a linear function).

The linear regression applet will open in a separate screen.

1.  Plot the points corresponding to the data in Table 1.  The first coordinate is year; denote this by t with t = 0 in year 2000.  The second coordinate is population in billions.

Enter the data provided in Table 1; enter the t value for each year (t = -10 for 1990, t = -5 for 1995, etc.) in the “x” column, then enter the population figures in the “y” column.

Click “Plot”.

2.    Determine the linear regression function that best fits these data; call this function P(t).  Graph the function P(t) on the same coordinate system as the plot of the data points.

Click “Analyze”. The screen shown below should appear.

You should obtain the linear function P(t) = 0.072t + 5.952.  Here, the variable x has been replaced by t, and the variable y has been replaced by the function notation P(t); coefficients are rounded to three places according to instructions.  The graph is shown on the applet screen.

3.         What is the Slope of P(t)?  Give a verbal interpretation of the answer; identify units clearly.

The slope is m = 0.072; this means that the world population is growing at the rate of 0.072 billion (72,000,000) people each year.  (There is not an applet for this!)

Use the function P(t) to answer the following questions.

4.         What is the annual population growth? 

The annual population growth is 72,000,000 people (see #3)

The next computations can be done using the "Java Math Pad" applet provided below. Usually, you will need to follow these steps.

Scroll down to “Tool Chest” on the menu to the left of the screen. 

Click “Java Math Pad”. This performs mathematical evaluations.

5.         How much will the population grow in 10 years? Six months? One week?

Type 10*72000000 then press enter. Your answer will appear in scientific notation,7.2E8 which is 720,000,000 people. Or, you can type 10*.072, press enter. Here your answer will be .72 billion people.

Type .5*72000000 then press enter.

Type (1/52)*72000000 then press enter.

In 10 years the population will grow by 10x72,000,000 = 720,000,000 people; in 6 months, 0.5x72,000,000 = 36,000,000 people; in one week, (1/52)x72,000,000 = 1,384,615 people. 

6.            Estimate the population in the year 2005.

In the year 2005, t = 5. You can continue using the “Java Math Pad”. 

Type in P(t) = 0.072*t + 5.952, then “Enter”.  Next type P(5), then “Enter”.  You should see the value of the function P at t = 5:

                        P(5) = 6.312 (rounded to 3 decimal places).

In the year 2005, there will be approximately 6.312 billion (6,312,000,000) people in the World.

7.            Predict when the population will reach 8,000,000,000.

To make this prediction, you can solve the linear equation

                        P(t) = 8. 

This can be done by hand algebraically fairly easily but we illustrate an applet that will do this graphically; this will make your work easier for more complicated equations.  The graphical solution will be the intersection of the graphs of P(t) and the horizontal line y = 8.

            Scroll down the menu to the “Tool Chest”, then click Plot-Solve. (First, read the "help" section.) Enter the first function P(t), then enter the second function as the constant 8. Click "plot". Then click on the screen to see a red point on the graph of one of the functions. Click on "right" or "left" to move the point to the intersection of the two graphs. You may want to zoom in for more accuracy. The coordinates of the point can be read on the screen.

In the year 2028 (t = 28.4), there will be approximately 8 billion people in the World.

Reflection

In this section, you should examine your model, solutions, and implications.  Think about the validity and reasonableness of your answers. How closely does your function fit the data?  Relevant questions are provided although you may raise questions of your own.  Record your answers to these questions and thoughts of your own in your journal.  

Questions

A.        Do you think that a linear function is good to use for this study?  Are there other functions that you think might provide a better model?  Why?

B.        How long do you think this model will be accurate; i.e., what is a reasonable domain for the function?

C.        How do you think the predicted increase in world population might affect future life in the World?  In the United States?  In your home town?

Now you try this on your own with the Practice Module