# 0.5 Mathematical models

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## Mathematical models - grade 11

Up until now, you have only learnt how to solve equations and inequalities, but there has not been much application of what you have learnt. This chapter builds on this knowledge and introduces you to the idea of a mathematical model which uses mathematical concepts to solve real-world problems.

Mathematical Model

A mathematical model is a method of using the mathematical language to describe the behaviour of a physical system. Mathematical models are used particularly in the natural sciences and engineering disciplines (such as physics, biology, and electrical engineering) but also in the social sciences (such as economics, sociology and political science); physicists, engineers, computer scientists, and economists use mathematical models most extensively.

A mathematical model is an equation (or a set of equations for the more difficult problems) that describes a particular situation. For example, if Anna receives R3 for each time she helps her mother wash the dishes and R5 for each time she helps her father cut the grass, how much money will Anna earn if she helps her mother 5 times to wash the dishes and helps her father 2 times to wash the car.The first step to modelling is to write the equation, that describes the situation. To calculate how much Anna will earn we see that she will earn :

$\begin{array}{ccc}& 5& ×R3\phantom{\rule{1.em}{0ex}}\mathrm{for washing the dishes}\hfill \\ \hfill +& 2& ×R5\phantom{\rule{1.em}{0ex}}\mathrm{for cutting the grass}\hfill \\ \hfill =& R15& +R10\hfill \\ \hfill =& R25\end{array}$

If however, we ask: "What is the equation if Anna helps her mother $x$ times and her father $y$ times?", then we have:

$\mathrm{Total earned}=\mathrm{x}×\mathrm{R}3+\mathrm{y}×\mathrm{R}5$

## Real-world applications: mathematical models

Some examples of where mathematical models are used in the real-world are:

1. To model population growth
2. To model effects of air pollution
3. To model effects of global warming
4. In computer games
5. In the sciences (e.g. physics, chemistry, biology) to understand how the natural world works
6. In simulators that are used to train people in certain jobs, like pilots, doctors and soldiers
7. In medicine to track the progress of a disease

## Investigation : simple models

In order to get used to the idea of mathematical models, try the following simple models. Write an equation thatdescribes the following real-world situations, mathematically:

1. Jack and Jill both have colds. Jack sneezes twice for each sneeze of Jill's. If Jill sneezes $x$ times, write an equation describing how many times they both sneezed?
2. It rains half as much in July as it does in December. If it rains $y$  mm in July, write an expression relating the rainfall in July and December.
3. Zane can paint a room in 4 hours. Billy can paint a room in 2 hours. How long will it take both of them to paint a room together?
4. 25 years ago, Arthur was 5 more than $\frac{1}{3}$ as old as Lee was. Today, Lee is 26 less than twice Arthur's age. How old is Lee?
5. Kevin has played a few games of ten-pin bowling. In the third game, Kevin scored 80 more than in the second game. In the first game Kevin scored 110 less than the third game. His total score for the first two games was 208. If he wants an average score of 146, what must he score on the fourth game?
6. Erica has decided to treat her friends to coffee at the Corner Coffee House. Erica paid R54,00 for four cups of cappuccino and three cups of filter coffee. If a cup of cappuccino costs R3,00 more than a cup of filter coffee, calculate how much each type of coffee costs?
7. The product of two integers is 95. Find the integers if their total is 24.

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
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
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
Santosh
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
Other chapter Q/A we can ask