Exponential Growth and Decay

One of the most prevalent applications of exponential functions involves growth and decay models. Exponential growth and decay show up in a host of natural applications. From population growth and continuously compounded interest to radioactive decay and Newton’s law of cooling, exponential functions are ubiquitous in nature. In this section, we examine exponential growth and decay in the context of some of these applications.

Exponential Growth Model

Many systems exhibit exponential growth. These systems follow a model of the form

y = y_{0} e^{k t},

is a positive constant, called the growth constant. Notice that in an exponential growth model, we have

That is, the rate of growth is proportional to the current function value. This is a key feature of exponential growth. [link] involves derivatives and is called a differential equation. We learn more about differential equations in Introduction to Differential Equations.

Population growth is a common example of exponential growth. Consider a population of bacteria, for instance. It seems plausible that the rate of population growth would be proportional to the size of the population. After all, the more bacteria there are to reproduce, the faster the population grows. [link] and [link] represent the growth of a population of bacteria with an initial population of

200

Note that we are using a continuous function to model what is inherently discrete behavior. At any given time, the real-world population contains a whole number of bacteria, although the model takes on noninteger values. When using exponential growth models, we must always be careful to interpret the function values in the context of the phenomenon we are modeling.

Exponential Growth of a Bacterial Population
Time (min)	Population Size (no. of bacteria)
$10$	$244$
$20$	$298$
$30$	$364$
$40$	$445$
$50$	$544$
$60$	$664$
$70$	$811$
$80$	$991$
$90$	$1210$
$100$	$1478$
$110$	$1805$
$120$	$2205$

Let’s now turn our attention to a financial application: compound interest. Interest that is not compounded is called simple interest. Simple interest is paid once, at the end of the specified time period (usually

1

Compound interest is paid multiple times per year, depending on the compounding period. Therefore, if the bank compounds the interest every

6

months. During the second half of the year, the account earns interest not only on the initial

$ 1000,

but also on the interest earned during the first half of the year. Mathematically speaking, at the end of the year, we have

If we extend this concept, so that the interest is compounded continuously, after

t

Now let’s manipulate this expression so that we have an exponential growth function. Recall that the number

e

Based on this, we want the expression inside the parentheses to have the form

(1 + 1 / m) .

Generalizing this concept, we see that if a bank account with an initial balance of

$ P

If a quantity grows exponentially, the time it takes for the quantity to double remains constant. In other words, it takes the same amount of time for a population of bacteria to grow from

100

bacteria. This time is called the doubling time. To calculate the doubling time, we want to know when the quantity reaches twice its original size. So we have

Exponential Decay Model

Exponential functions can also be used to model populations that shrink (from disease, for example), or chemical compounds that break down over time. We say that such systems exhibit exponential decay, rather than exponential growth. The model is nearly the same, except there is a negative sign in the exponent. Thus, for some positive constant

k,

As with exponential growth, there is a differential equation associated with exponential decay. We have

The following figure shows a graph of a representative exponential decay function.

Let’s look at a physical application of exponential decay. Newton’s law of cooling says that an object cools at a rate proportional to the difference between the temperature of the object and the temperature of the surroundings. In other words, if

T

Note that this is not quite the right model for exponential decay. We want the derivative to be proportional to the function, and this expression has the additional

T_{a}

term. Fortunately, we can make a change of variables that resolves this issue. Let

y (t) = T (t) - T_{a} .

From our previous work, we know this relationship between y and its derivative leads to exponential decay. Thus,

represents the initial temperature. Let’s apply this formula in the following example.

Just as systems exhibiting exponential growth have a constant doubling time, systems exhibiting exponential decay have a constant half-life. To calculate the half-life, we want to know when the quantity reaches half its original size. Therefore, we have

Radiocarbon Dating

One of the most common applications of an exponential decay model is carbon dating. $Carbon- 14$

decays (emits a radioactive particle) at a regular and consistent exponential rate. Therefore, if we know how much carbon was originally present in an object and how much carbon remains, we can determine the age of the object. The half-life of $carbon- 14$

is approximately $5730$

years—meaning, after that many years, half the material has converted from the original $carbon- 14$

to the new nonradioactive $nitrogen- 14 .$

If we have $100$

g $carbon- 14$

today, how much is left in $50$

years? If an artifact that originally contained $100$

g of carbon now contains $10$

g of carbon, how old is it? Round the answer to the nearest hundred years.

We have

\begin{matrix} 5730 & = & \frac{ln 2}{k} \\ k & = & \frac{ln 2}{5730} . \end{matrix}

So, the model says

y = 100 e^{− (ln 2 / 5730) t} .

In $50$

years, we have

\begin{matrix} y & = & 100 e^{− (ln 2 / 5730) (50)} \\ \approx & 99.40 . \end{matrix}

Therefore, in $50$

years, $99.40$

g of $carbon- 14$

remains.

To determine the age of the artifact, we must solve

\begin{matrix} 10 & = & 100 e^{− (ln 2 / 5730) t} \\ \frac{1}{10} & = & e^{− (ln 2 / 5730) t} \\ t & \approx & 19035. \end{matrix}

The artifact is about $19,000$

years old.

Key Concepts

True or False? If true, prove it. If false, find the true answer.

The doubling time for $y = e^{c t}$

is $(ln (2)) / (ln (c)) .$

If you invest $$ 500,$

an annual rate of interest of $3 %$

yields more money in the first year than a $2.5 %$

continuous rate of interest.

True

If you leave a $100 ° C$

pot of tea at room temperature $(25 ° C)$

and an identical pot in the refrigerator $(5 ° C),$

with $k = 0.02,$

the tea in the refrigerator reaches a drinkable temperature $(70 ° C)$

more than $5$

minutes before the tea at room temperature.

If given a half-life of t years, the constant $k$

for $y = e^{k t}$

is calculated by $k = ln (1 / 2) / t .$

False; $k = \frac{ln (2)}{t}$

For the following exercises, use $y = y_{0} e^{k t} .$

If a culture of bacteria doubles in $3$

hours, how many hours does it take to multiply by $10 ?$

If bacteria increase by a factor of $10$

in $10$

hours, how many hours does it take to increase by $100 ?$

20

hours

How old is a skull that contains one-fifth as much radiocarbon as a modern skull? Note that the half-life of radiocarbon is $5730$

years.

If a relic contains $90 %$

as much radiocarbon as new material, can it have come from the time of Christ (approximately $2000$

years ago)? Note that the half-life of radiocarbon is $5730$

years.

No. The relic is approximately $871$

years old.

The population of Cairo grew from $5$

million to $10$

million in $20$

years. Use an exponential model to find when the population was $8$

million.

The populations of New York and Los Angeles are growing at $1 %$

and $1.4 %$

a year, respectively. Starting from $8$

million (New York) and $6$

million (Los Angeles), when are the populations equal?

71.92

years

Suppose the value of $$ 1$

in Japanese yen decreases at $2 %$

per year. Starting from $$ 1 = ¥ 250,$

when will $$ 1 = ¥ 1 ?$

The effect of advertising decays exponentially. If $40 %$

of the population remembers a new product after $3$

days, how long will $20 %$

remember it?

5

days $6$

hours $27$

minutes

If $y = 1000$

at $t = 3$

and $y = 3000$

at $t = 4,$

what was $y_{0}$

at $t = 0 ?$

If $y = 100$

at $t = 4$

and $y = 10$

at $t = 8,$

when does $y = 1 ?$

12

If a bank offers annual interest of $7.5 %$

or continuous interest of $7.25 %,$

which has a better annual yield?

What continuous interest rate has the same yield as an annual rate of $9 % ?$

8.618 %

If you deposit $$ 5000$

at $8 %$

annual interest, how many years can you withdraw $$ 500$

(starting after the first year) without running out of money?

You are trying to save $$ 50,000$

in $20$

years for college tuition for your child. If interest is a continuous $10 %,$

how much do you need to invest initially?

$ 6766.76

You are cooling a turkey that was taken out of the oven with an internal temperature of $165 ° F .$

After $10$

minutes of resting the turkey in a $70 ° F$

apartment, the temperature has reached $155 ° F .$

What is the temperature of the turkey $20$

minutes after taking it out of the oven?

You are trying to thaw some vegetables that are at a temperature of $1 ° F .$

To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of $44 ° F .$

You check on your vegetables $2$

hours after putting them in the refrigerator to find that they are now $12 ° F .$

Plot the resulting temperature curve and use it to determine when the vegetables reach $33 ° F .$

9

hours $13$

minutes

You are an archaeologist and are given a bone that is claimed to be from a Tyrannosaurus Rex. You know these dinosaurs lived during the Cretaceous Era $(146$

million years to $65$

million years ago), and you find by radiocarbon dating that there is $0.000001 %$

the amount of radiocarbon. Is this bone from the Cretaceous?

The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of $24,000$

years. If $1$

barrel containing $10 kg$

of plutonium-239 is sealed, how many years must pass until only $10 g$

of plutonium-239 is left?

239,179

years

For the next set of exercises, use the following table, which features the world population by decade.

*Source*: http://www.factmonster.com/ipka/A0762181.html.
Years since 1950	Population (millions)
$0$	$2,556$
$10$	$3,039$
$20$	$3,706$
$30$	$4,453$
$40$	$5,279$
$50$	$6,083$
$60$	$6,849$

[T] The best-fit exponential curve to the data of the form $P (t) = a e^{b t}$

is given by $P (t) = 2686 e^{0.01604 t} .$

Use a graphing calculator to graph the data and the exponential curve together.

[T] Find and graph the derivative $y^{'}$

of your equation. Where is it increasing and what is the meaning of this increase?

P' (t) = 43 e^{0.01604 t} .

The population is always increasing.

[T] Find and graph the second derivative of your equation. Where is it increasing and what is the meaning of this increase?

[T] Find the predicted date when the population reaches $10$

billion. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.

The population reaches $10$

billion people in $2027 .$

For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century.

*Source*: http://www.sfgenealogy.com/sf/history/hgpop.htm.
Years since 1850	Population (thousands)
$0$	$21.00$
$10$	$56.80$
$20$	$149.5$
$30$	$234.0$

[T] The best-fit exponential curve to the data of the form $P (t) = a e^{b t}$

is given by $P (t) = 35.26 e^{0.06407 t} .$