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Exponential Growth and Decay

28 April 2014

Exponential Growth and Decay 28 April 2014 1/24

This week we’ll talk about a few situations which behavemathematically like compound interest. They include populationgrowth, radioactive decay, and google map zooming.

To help make the connection with interest rates, we discuss one moreidea of compound interest.


If you have money in a savings account, then the amount of interestyou make per period is proportional to the amount of money in theaccount. The constant ratio of interest to principal is the interest ratepaid to you.

A consequence of this proportionality is that if you double how muchmoney you have in the account, you’ll double the interest you get.


Continuous Compounding

Let’s suppose we invest $100 in a savings account paying 5% peryear. 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% perquarter. There are 4 quarters in a year, so after one year we’d have

$100 · (1 + .05/4)4 = $105.09

If interest is compounded monthly, after one year we’d have

$100 · (1 + .05/12)12 = $105.12


If we compound daily, after one year we’d have

$100 · (1 + .05/365)365 = $105.13

If we compound hourly, after one year we’d have

$100 · (1 + .05/8760)8760 = $105.13

By compounding more and more frequently, we are increasing howmuch we get, but only by a little.

The most extreme notion is called compounding continuously. Thename gives a rough idea of what this means.


There is a number, usually denoted by e, whose value isapproximately 2.7, which helps to calculate continuous compounding.If you invest P0 in an account paying r% per year, compoundedcontinuously.after t years, the amount of money you’ll have is

P0 · ert

On a calculator, or a spreadsheet, the function ex is often writtenexp(x). Some calculators will have an exp button, and some will havea button for the number e.


When a quantity grows at a rate proportional to its size, then theequation

P = P0 · ert

governs the size, where P0 represents the size at some initial time, ris the growth rate, and t is the amount of time past the initial time.

The situation of something growing proportionally to its size occurs ina number of situations. When this happens the quantity is said togrow exponentially.


For example, we can apply this idea to compound interest, where P0

is our initial deposit, r is our annual interest rate (compoundingcontinuously) and t is the number of years we’ve left the money inthe bank.

For example, if we deposit $100 in the bank paying 5% compoundedcontinuously, after 1 year we’ll have

$100 · e0.05·1 = $100 · e0.05 = $105.13

Compounding 5% continuously is equivalent to 5.13% compoundedyearly.


Clicker Question

If you deposit $100 in a savings account which pays an annual rate of6% interest, compounded continuously, how much money will youhave in 3 years?

To enter the expression P0 · ert on a calculator, the following stepsmost likely will work

P0 × e ∧ ( r × t ) =

orP0 × e xy ( r × t ) =


Answer

The amount you’ll have in 3 years is

$100 · e0.06·3 = $100 · e0.18 = $119.72

The webpage http://ultimatecalculators.com does both standardcompound interest and continuous compounding.


http://www.ultimatecalculators.com/continuous_compounding_calculator.html

Population Growth

A first assumption about population growth is often that the rate ofbirths and deaths is proportional to the size of the population. Thatis, if the population doubles, then the number of births and deathseach double. The growth rate is then proportional to the size of thepopulation.

This is a reasonable assumption for many populations, includinghuman populations, although it can be simplistic, especially whenresources are scarce.

Assuming this model, population would be governed by the equation

P = P0 · ert

where P0 is the population at some given time, r is the annualpopulation growth rate, and t is the number of years past the giventime.


An Example

In 1990 the world population was estimated to be 5.3 billion andincreasing at the rate of 1.7% per year. What would the worldpopulation be in 2000?

We can view the initial population (in 1990) as 5.3 (measured inbillions). If t is the number of years past 1990, then the worldpopulation, if it grows at 1.7% per year, would be given by theequation

P = 5.3 · e .017t

In 2000, 10 years would have passed since 1990, so the populationwould be estimated as

5.3 · e .017·10 = 5.3 · e .17 = 6.28

or about 6.3 billion people. The actual population, according to theU.S. Census Bureau, was about 6.1 billion.


Population Doubling Time

Because the equation governing population growth is the same as forcompound interest, some of the same consequences happen forpopulations.

We saw that the doubling time for compound interest only dependedon the interest rate and not on the amount of money we deposit.

The same thing happens for population growth. That is, the time ittakes for a population to double doesn’t depend on the initialpopulation.


Clicker Question

Approximately how long would it take for a population to double if itis increasing at the rate of 1.7% per year?

Recall that we had a doubling rule of thumb, for compound interest,that said

doubling time =72

interest rate

If we use the percentage, rather than the decimal, for the interest rate.


Answer

The rule of thumb would give us

doubling time =72

1.7= 42.3 years

or a little over 42 years.


There will be some situations we’ll look at where we need somethingmore detailed than this rule of thumb.

The actual equation we’d want to solve to answer how long it takesfor a population to double when it is increasing at a rate of 1.7% peryear is

P0 · e .017t = 2P0

Dividing by P0, it amounts to asking for the value of t for which

e0.17t = 2


When you want to solve the equation e0.17t = 2, where the unknownis in the exponent, you use logarithms.

Scientific calculators usually have a button named ln. The meaning ofit is, if x is a number with ex = 2, then x = ln(2).

For another example, if ex = 7.1, then x = ln(7.1).

The symbol ln stands for natural logarithm. Using this operationallows us to solve equations where the unknown is in the exponent.


Getting back to the doubling question, if the population increases1.7% per year, we obtained the equation

e0.017t = 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)

0.017= 40.8 years


In general, if r is the population growth rate, then solving to find thedoubling time, we’d get

doubling time =ln(2)

r

In fact, the doubling rule of thumb we gave last week is anapproximation to this more complicated formula.


Another Example

An advertisement for Paul Kennedy’s book Preparing for theTwenty-First Century (Random House, 1993) asks: “By 2025, Africa’spopulation will be: 50%, 150%, or 300% greater than Europe’s?” Thepopulation of Europe in mid-1993 was 500 million and was expectedto stay constant through 2025. The population of Africa in mid-1993was 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 millionin 1993, and if it grows at 2.9% per year, what will it be in 2025?


Answer

We have P0 = 720 million and r = .029. The year 2025 is 32 yearsafter 1993, so t = 32. The population of Africa in 2025 can then beestimated by

720 · e0.029·32 = 720 · e0.928 = 1821 million

or about 1.8 billion people. Compared to Europe’s estimatedpopulation of 500 million, this is over three times as much, so theanswer to Kennedy’s question would be 300%.

As of 2010, Africa’s population is approximately 143% of Europe’spopulation.


The assumption that population grows exponentially typically is validonly when resources are abundant. For example, if a population ofpredators begins to decimate its prey, then the population will ceaseto grow exponentially, and can begin to decrease if the amount ofprey is low enough.

Human population is fairly poorly estimated by this model, in largepart because the growth rate varies over time. If one is interested insmall enough time intervals, where the growth rate is fairly constant,then this model is pretty good.

The website www.indexmundi.com uses CIA data to show annualgrowth rates of countries over about a 10 year period.


http://www.indexmundi.com/g/g.aspx?c=us&v=24

Next Time

We’ll start with radioactive decay on Wednesday.


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