# 2.8 Exponential growth and decay

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• Use the exponential growth model in applications, including population growth and compound interest.
• Explain the concept of doubling time.
• Use the exponential decay model in applications, including radioactive decay and Newton’s law of cooling.
• Explain the concept of half-life.

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 $y={y}_{0}{e}^{kt},$ where ${y}_{0}$ represents the initial state of the system and $k$ is a positive constant, called the growth constant . Notice that in an exponential growth model, we have

${y}^{\prime }=k{y}_{0}{e}^{kt}=ky.$

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 .

## Rule: exponential growth model

Systems that exhibit exponential growth    increase according to the mathematical model

$y={y}_{0}{e}^{kt},$

where ${y}_{0}$ represents the initial state of the system and $k>0$ is a constant, called the growth constant .

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 $200$ bacteria and a growth constant of $0.02.$ Notice that after only $2$ hours $\left(120$ minutes), the population is $10$ times its original size!

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

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.

## Population growth

Consider the population of bacteria described earlier. This population grows according to the function $f\left(t\right)=200{e}^{0.02t},$ where t is measured in minutes. How many bacteria are present in the population after $5$ hours $\left(300$ minutes)? When does the population reach $100,000$ bacteria?

We have $f\left(t\right)=200{e}^{0.02t}.$ Then

$f\left(300\right)=200{e}^{0.02\left(300\right)}\approx 80,686.$

There are $80,686$ bacteria in the population after $5$ hours.

To find when the population reaches $100,000$ bacteria, we solve the equation

$\begin{array}{ccc}\hfill 100,000& =\hfill & 200{e}^{0.02t}\hfill \\ \hfill 500& =\hfill & {e}^{0.02t}\hfill \\ \hfill \text{ln}\phantom{\rule{0.2em}{0ex}}500& =\hfill & 0.02t\hfill \\ \hfill t& =\hfill & \frac{\text{ln}\phantom{\rule{0.2em}{0ex}}500}{0.02}\approx 310.73.\hfill \end{array}$

The population reaches $100,000$ bacteria after $310.73$ minutes.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
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