Exponential Growth and Decay

Transcription

1 Exponential Growth and Decay 28 April 2014 Exponential Growth and Decay 28 April /24

2 This week we ll talk about a few situations which behave mathematically like compound interest. They include population growth, radioactive decay, and google map zooming. To help make the connection with interest rates, we discuss one more idea of compound interest. Exponential Growth and Decay 28 April /24

3 If you have money in a savings account, then the amount of interest you make per period is proportional to the amount of money in the account. The constant ratio of interest to principal is the interest rate paid to you. A consequence of this proportionality is that if you double how much money you have in the account, you ll double the interest you get. Exponential Growth and Decay 28 April /24

4 Continuous Compounding Let s suppose we invest $100 in a savings account paying 5% per year. If the interest is compounded yearly, after one year we d have $100 (1 +.05) = $105. If interest is compounded quarterly, then we get 5%/4 = 1.25% per quarter. There are 4 quarters in a year, so after one year we d have $100 (1 +.05/4) 4 = $ If interest is compounded monthly, after one year we d have $100 (1 +.05/12) 12 = $ Exponential Growth and Decay 28 April /24

5 If we compound daily, after one year we d have $100 (1 +.05/365) 365 = $ If we compound hourly, after one year we d have $100 (1 +.05/8760) 8760 = $ By compounding more and more frequently, we are increasing how much we get, but only by a little. The most extreme notion is called compounding continuously. The name gives a rough idea of what this means. Exponential Growth and Decay 28 April /24

6 There is a number, usually denoted by e, whose value is approximately 2.7, which helps to calculate continuous compounding. If you invest P 0 in an account paying r% per year, compounded continuously.after t years, the amount of money you ll have is P 0 e rt On a calculator, or a spreadsheet, the function e x is often written exp(x). Some calculators will have an exp button, and some will have a button for the number e. Exponential Growth and Decay 28 April /24

7 When a quantity grows at a rate proportional to its size, then the equation P = P 0 e rt governs the size, where P 0 represents the size at some initial time, r is the growth rate, and t is the amount of time past the initial time. The situation of something growing proportionally to its size occurs in a number of situations. When this happens the quantity is said to grow exponentially. Exponential Growth and Decay 28 April /24

8 For example, we can apply this idea to compound interest, where P 0 is our initial deposit, r is our annual interest rate (compounding continuously) and t is the number of years we ve left the money in the bank. For example, if we deposit $100 in the bank paying 5% compounded continuously, after 1 year we ll have $100 e = $100 e 0.05 = $ Compounding 5% continuously is equivalent to 5.13% compounded yearly. Exponential Growth and Decay 28 April /24

9 Clicker Question If you deposit $100 in a savings account which pays an annual rate of 6% interest, compounded continuously, how much money will you have in 3 years? To enter the expression P 0 e rt on a calculator, the following steps most likely will work P 0 e ( r t ) = or P 0 e x y ( r t ) = Exponential Growth and Decay 28 April /24

10 Answer The amount you ll have in 3 years is $100 e = $100 e 0.18 = $ The webpage does both standard compound interest and continuous compounding. Exponential Growth and Decay 28 April /24

11 Population Growth A first assumption about population growth is often that the rate of births and deaths is proportional to the size of the population. That is, if the population doubles, then the number of births and deaths each double. The growth rate is then proportional to the size of the population. This is a reasonable assumption for many populations, including human populations, although it can be simplistic, especially when resources are scarce. Assuming this model, population would be governed by the equation P = P 0 e rt where P 0 is the population at some given time, r is the annual population growth rate, and t is the number of years past the given time. Exponential Growth and Decay 28 April /24

12 An Example In 1990 the world population was estimated to be 5.3 billion and increasing at the rate of 1.7% per year. What would the world population be in 2000? We can view the initial population (in 1990) as 5.3 (measured in billions). If t is the number of years past 1990, then the world population, if it grows at 1.7% per year, would be given by the equation P = 5.3 e.017t In 2000, 10 years would have passed since 1990, so the population would be estimated as 5.3 e = 5.3 e.17 = 6.28 or about 6.3 billion people. The actual population, according to the U.S. Census Bureau, was about 6.1 billion. Exponential Growth and Decay 28 April /24

13 Population Doubling Time Because the equation governing population growth is the same as for compound interest, some of the same consequences happen for populations. We saw that the doubling time for compound interest only depended on the interest rate and not on the amount of money we deposit. The same thing happens for population growth. That is, the time it takes for a population to double doesn t depend on the initial population. Exponential Growth and Decay 28 April /24

14 Clicker Question Approximately how long would it take for a population to double if it is increasing at the rate of 1.7% per year? Recall that we had a doubling rule of thumb, for compound interest, that said 72 doubling time = interest rate If we use the percentage, rather than the decimal, for the interest rate. Exponential Growth and Decay 28 April /24

15 Answer The rule of thumb would give us or a little over 42 years. doubling time = 72 = 42.3 years 1.7 Exponential Growth and Decay 28 April /24

16 There will be some situations we ll look at where we need something more detailed than this rule of thumb. The actual equation we d want to solve to answer how long it takes for a population to double when it is increasing at a rate of 1.7% per year is P 0 e.017t = 2P 0 Dividing by P 0, it amounts to asking for the value of t for which e 0.17t = 2 Exponential Growth and Decay 28 April /24

17 When you want to solve the equation e 0.17t = 2, where the unknown is in the exponent, you use logarithms. Scientific calculators usually have a button named ln. The meaning of it is, if x is a number with e x = 2, then x = ln(2). For another example, if e x = 7.1, then x = ln(7.1). The symbol ln stands for natural logarithm. Using this operation allows us to solve equations where the unknown is in the exponent. Exponential Growth and Decay 28 April /24

18 Getting back to the doubling question, if the population increases 1.7% per year, we obtained the equation e 0.017t = 2 and finding t would give the doubling time. Taking logarithms gives 0.017t = ln(2) so we can solve for t by dividing by 0.017, getting t = ln(2) = 40.8 years Exponential Growth and Decay 28 April /24

19 In general, if r is the population growth rate, then solving to find the doubling time, we d get doubling time = ln(2) r In fact, the doubling rule of thumb we gave last week is an approximation to this more complicated formula. Exponential Growth and Decay 28 April /24

20 Another Example An advertisement for Paul Kennedy s book Preparing for the Twenty-First Century (Random House, 1993) asks: By 2025, Africa s population will be: 50%, 150%, or 300% greater than Europe s? The population of Europe in mid-1993 was 500 million and was expected to stay constant through The population of Africa in mid-1993 was 720 million and was increasing at about 2.9% per year. What answer would we give to Kennedy s question? Essentially, what he is asking is if Africa s population was 720 million in 1993, and if it grows at 2.9% per year, what will it be in 2025? Exponential Growth and Decay 28 April /24

21 Answer We have P 0 = 720 million and r =.029. The year 2025 is 32 years after 1993, so t = 32. The population of Africa in 2025 can then be estimated by 720 e = 720 e = 1821 million or about 1.8 billion people. Compared to Europe s estimated population of 500 million, this is over three times as much, so the answer to Kennedy s question would be 300%. As of 2010, Africa s population is approximately 143% of Europe s population. Exponential Growth and Decay 28 April /24

22 The assumption that population grows exponentially typically is valid only when resources are abundant. For example, if a population of predators begins to decimate its prey, then the population will cease to grow exponentially, and can begin to decrease if the amount of prey is low enough. Human population is fairly poorly estimated by this model, in large part because the growth rate varies over time. If one is interested in small enough time intervals, where the growth rate is fairly constant, then this model is pretty good. The website uses CIA data to show annual growth rates of countries over about a 10 year period. Exponential Growth and Decay 28 April /24

23 Next Time We ll start with radioactive decay on Wednesday. Exponential Growth and Decay 28 April /24