0.5 Mathematical models

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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.

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