In the previous section, we completed the examples of mathematical modelling. We saw four examples altogether. Those examples helped us to learn the basics about mathematical modelling. In this section, we will see some interesting situations where mathematical modelling can be effectively used.
1. In human beings and all other animals, blood flows through blood vessels and reach all parts of the body. Blood carries the oxygen and nutrients which must reach all parts. Any obstruction to the flow can cause serious health issues. The characteristics of the blood vessels are such that there is no obstruction to the flow. But due to some illness, those characteristics can change. In such a situation, a mathematical model can be prepared. The model will relate the new characteristics and the resulting new pattern of blood flow. Based on the results given by the mathematical model, the physiologist can make better judgements.
2. In cricket, the third umpire has to take decisions about LBW. He analyses the trajectory of the ball assuming that the batsman is not present. A mathematical model of the trajectory will help the umpire to decide whether an LBW will occur if the batsman was actually present.
3. Meteorology department has to make whether predictions. The parameters that they have to consider are: temperature, air pressure, humidity, wind speed etc., A mathematical model is prepared by considering all the essential parameters. Precision instruments are available to measure these parameters. Using the values obtained from the instruments, the mathematical model will give the required results. Interpretation of the results enable the department to make predictions about the whether conditions.
4. Department of Agriculture can predict the quantity of rice that can be harvested in a year. The quantity of rice available from a unit area is an essential parameter. Fertility of the soil in different parts of the country, availability of water for proper irrigation etc., are also essential parameters. The mathematical model is prepared based on the principles of statistics. Interpretation of the results given by the model, enable the department to make predictions about the quantity of rice.
Given below is an actual situation which occurred in the 18th century. It was solved by mathematical modelling. It can be written in 3 steps:
1. The Pregel river flows through Konigsberg town. So the town is on the two river banks. River bank A and River bank B. This is shown in the fig.B.4 below:
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| Fig.B.4 |
• There are two islands C and D in the river. Seven bridges connect the islands and the banks.
2. The problem is to go around the town, visiting both the islands.
• But three conditions must be satisfied:
(i) The person can start the journey from any river bank or any island. But the journey must end at the exact starting point.
(ii) The person must use all the bridges.
(iii) Each bridge must be used only once.
3. Both residents and authorities could not solve the problem. So they requested the famous mathematician Leonhard Euler to find a solution.
• The solution put forward by Euler can be written in 5 steps:
(i) In a pictorial representation of the problem, bridges are denoted by arcs or lines. The end points of bridges are denoted by dots. Those dots are called vertices (plural of vertex).
(ii) Let us see some examples:
In fig.B.5(I) below,
• Three arcs start from A. This is because, there are three bridges starting from the river bank A. Two of them to island C and the third to island D.
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| Fig.B.5 |
• Four arcs and one line start from C. This is because, there are five bridges starting from the island C. Two of them to river bank A, two of them to river bank B and the fifth to island D.
(iii) We see that:
♦ A has 3 arcs/lines. 3 is an odd number.
♦ B has 3 arcs/lines. 3 is an odd number.
♦ C has 5 arcs/lines. 5 is an odd number.
♦ D has 3 arcs/lines. 3 is an odd number.
• If a vertex has an odd number of arcs/lines, that vertex is called an odd vertex.
• If a vertex has an even number of arcs/lines, that vertex is called an even vertex.
• So in this problem, all are odd vertices.
(iv) Euler said that, in a journey which has to satisfy the conditions in (2), there must be only two odd vertices.
• In the present problem, there are four odd vertices. So will never be able to satisfy the conditions.
(v) Later, a new bridge was added between A and B. This is shown in fig.B.5(II). Now C and D are the required two odd vertices while A and B are even vertices.
We have completed a discussion on mathematical modelling. With that, we have completed all the chapters of class 11. In the next chapter, we will see the first chapter of class 12.
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