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

• 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.

•  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

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.

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.

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

How to determine the depth of the river?

• 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, \(x \text{ m}\).

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.

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

• 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.

•  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

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.

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.

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

How to determine the depth of the river?

• 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, \(x \text{ m}\).

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.