## Mathematical Modeling

 8.1 Mathematical Modeling

 What is mathematical modeling? A mathematical model is a representation of a system or scenario that is used to gain qualitative and/or quantitative understanding of some real-world problems and to predic future behaviour.

Components in mathematical modeling:

1. Identifying and defining the problems
2. Making assumptions and identifying the variables
3. Applying mathematics to solve problems
4. Verifying and interpreting solutions in the context of the problem
5. Refining the mathematical model
6. Reporting the findings

 Example 1 Problem Amin travelled 405 km on 45 litres of petrol in his car. If Amin wishes to go to a place which is 198 km away by car, how much petrol, in litres, does he need? Solve this problem through mathematical modelling. Identifying and defining the problem Determine the volume of petrol required for 198 km. We know that the farther we travel, the more petrol we require. Thus, the amount of petrol varies directly with the distance we travelled. Making assumptions and identifying the variables Assume that the driving speed for both 405 km and 198 km routes are the same Let $$x$$ represents the distance travelled and $$y$$ represents the amount of petrol required $$y$$ varies directly with $$x$$, hence $$y=kx$$ where $$k$$ is a constant

 Applying mathematics to solve problems $$\text{Substitute}\\ y=45\text{ and } x=405 \\ \text{ into } y=kx\\ 45=k(405) \implies k=\dfrac{45}{405}=\dfrac{1}{9}\\ \therefore y=\dfrac{1}{9}x\\$$ This equation describes the relationship between the amount of petrol required and the distance travelled. $$\text{When } x = 198,\\ y = \dfrac{1}{9}(198)= 22 \text{ litres}$$ Hence, 22 litres of petrol is required to travel 198 km.

 Verifying and interpreting solutions in the context of the problem We may not be able to use the linear function model \begin{aligned} y=\frac{1}{9}x \end{aligned} in all situations. For example, if the 405 km route is through towns and cities, and the 198 km route is through highway. Thus, the car will use up petrol at a faster rate in the first route compared to the second route. When this is translated to the real-world situation, the linear function model obtained is not suitable to solve this problem. Refining the mathematical model n this problem, we are not able to refine the model due to the limited information given. Reporting the findings Report the findings of the problem solving based on the interpretation of solutions as shown in the preceding sections.

Example 2
Problem

The diagram below shows the cross-section of a river. A hydrologist measures the depth of the river, $$y \text{ m}$$, at different distance, $$x \text{ m}$$ from the riverbank. The results obtained are given in the following table.

 Distance from the riverbank, $$x \text{ m}$$ Depth of the river, $$y \text{ m}$$ 0 0 4 1.5 8 2.3 12 2.9 18 2.9 25 1.7 30 0

Show how the hydrologist uses the data above to determine the depth of the river through mathematical modelling.

Identifying and defining the problem

How to determine the depth of the river?

Making assumptions and identifying the variables
• Assume the river is the deepest in the middle with the depth decreasing to 0 at the edges.
• The two variables involved in this study are the depth of the river, $$y \text{ m}$$, and the distance from the riverbank, $$x \text{ m}$$.

 Applying mathematics to solve problems Write the distance from the riverbank and the depth of the river as a set of ordered pairs $$(x, y)$$ and draw a graph for the data. The data seem to rise and fall in a manner similar to a quadratic function. The graph drawn shows the curve of best fit and resembles the graph of a quadratic function. In mathematical modeling to represent the actual situation, the approximate value is used. Based on the graph, the depth of the river is 3 m when the distance from the riverbank is 15 m (approximate).

 Verifying and interpreting solutions in the context of the problem Determine the related quadratic function of the form $$y = ax^2 + bx + c$$. Determine the constants $$a, b \text{ and } c$$ by substituting any three data, for example $$(0, 0), (25, 1.7) \text{ and } (30, 0)$$ into the equation.  $$\text{(0,0)}\\ 0 = a(0)^2 + b(0) + c\\ c=0\\ \, \\ \text{(25,1.7)}\\ 1.7 = a(25)^2 + b(25) + c\\1.7 = 625a + 25b + c\\ \,\\ \text{(30,0)}\\ 0 = a(30)^2 + b(30) + c\\ 0 = 900a + 30b + c$$ Since $$c = 0$$, the system of two linear equations in two variables is;  \begin{aligned} &1.7 = 625a + 25b \dots (1)\\ &\hspace{3.5mm}0 = 900a + 30b \dots(2)\\\\ &\text{From (2), } b=-30a\dots(3) \end{aligned}   Substitute (3) into (1), we obtain $$a-0.0136$$. Substitute $$a-0.0136$$ into (3), we obtain $$b=0.408$$.    $$\therefore y=−0.0136x^2 + 0.408x\dots(3)$$ Substitute $$x=15$$ into (3), we obtain $$y=3.06$$ Refining the mathematical model For this model, we assume that the river is the deepest in the middle. This may not be true for some other rivers. A new model will be needed if we have new assumptions. The answer will be more accurate if more data have been collected. Reporting the findings Write a full report following the above modelling framework structure.