In the previous section, we completed a discussion on infinite series. In this appendix B, we will see mathematical modelling.
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Many difficult situations that we face in science, engineering, economics, finance, management, etc., can be written in the form of mathematical problems.
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When we write such mathematical problems, there are two advantages:
♦ A better understanding of the situation can be obtained.
♦ Better solutions can be obtained to overcome the situation.
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Nowadays it has become a trend to translate difficult situations into mathematical problems. This is partly due to the fact that, advanced digital computers are now available to solve lengthy mathematical problems.
• The process of translating a situation into a mathematical problem is called mathematical modeling. We can become experts in mathematical modeling by practicing different types of problems. Let us see a few examples:
Example 1:
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The situation is:
A surveyor wants to measure the height of a tower. Since the tower is very tall, a measuring tape cannot be used.
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This situation can be converted into a mathematical problem. For that, we engage in the process of mathematical modeling. The final result can be obtained in 4 steps:
Step 1:
The surveyor studies the situation. He realizes that, a triangle can be drawn and trigonometry can be applied. For applying trigonometry, he will need three parameters:
(i) Height of the tower (h)
(ii) Angle of elevation (𝜃)
(iii) Distance from the foot of the tower to the point at which angle of elevation is measured (d).
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We can write:
Step 1 involves studying the situation and identifying the parameters.
Once identified, the surveyor proceeds to obtain those parameters which can be measured.
♦ He measures d using a tape measure. It is 450 m
♦ He measures 𝜃 using a theodolite. It is 40o.
Step 2:
The surveyor draws the triangle and writes the relevant mathematical equations. The triangle is shown in fig.B.1 below:
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| Fig.B.1 |
♦ Length AB is h
♦ Length OA is d
♦ Angle AOB is 𝜃.
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The equation connecting the parameters is:
$\begin{array}{ll}{} &{\tan \theta} & {~=~} &{\frac{AB}{OA}} &{} \\
{\Rightarrow} &{\tan \theta} & {~=~} &{\frac{h}{d}} &{} \\
{\Rightarrow} &{h} & {~=~} &{d \tan \theta} &{} \\
{\Rightarrow} &{h} & {~=~} &{450 \times \tan 40} &{} \\
\end{array}$
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We can write:
Step 2 involves drawing the necessary diagrams and writing the relevant mathematical equations. In short, we write a mathematical problem in this step. Any mathematical problem will have a definite solution. So in this step, it is necessary to recheck all the works done thus far.
Step 3:
The surveyor solves the equation. He gets h = 377.6 m
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We can write:
Step 3 involves solving the mathematical problem obtained in step 2. Calculators or digital computers can be used for lengthy problems.
Step 4:
The surveyor writes the height of the tower:
The height of the tower is 378 m.
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We can write:
In step 4, we write the result of the mathematical calculations. With this final step, the situation is overcome.
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This final step is called:
Interpreting the mathematical solution to the real solution.
In the next section, we will see the case of simple pendulum.
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