Chapter 8 — Appendix 2: Mathematical Modelling

The appendix defines mathematical modelling as an activity in which models are made to describe the behaviour of various phenomena using words, drawings, sketches, computer programs, or mathematical formulae — essentially converting a physical situation into mathematics using suitable conditions.

Step 1: Identify the physical situation. Step 2: Convert it into a mathematical model by introducing parameters/variables and applying known physical laws. Step 3: Find the solution of the mathematical problem. Step 4: Interpret the result and compare with observations or experiments. Step 5: If there is good agreement, accept the model; otherwise revise the assumptions and return to Step 2.

The appendix lists: (i) identify the need for the model, (ii) list required parameters/variables, (iii) identify available relevant data, (iv) identify applicable circumstances/assumptions, (v) identify governing physical principles, (vi) identify the equations, calculations, and solution approach, (vii) identify tests for consistency and utility of the model, and (viii) identify parameter values that can improve the model.

Let H be the tower height, h the observer's eye height, l the horizontal distance, and α the angle of elevation to the top. The model gives H = h + l tan α. If the tower foot is inaccessible (l unknown), an angle of depression β to the foot yields l = h cot β, removing the unknown.

The raw-material constraints for products P1, P2, P3 are written as a 3×3 linear system (3x + 7y + 5z = 330; 4x + 9y + 12z = 455; 3y + 7z = 140) and expressed in matrix form. Elementary row operations reduce the coefficient matrix to the identity, giving the solution x = 20, y = 35, z = 5, making full use of all available raw material.

With fixed cost a, variable cost b per unit, selling price s per unit, and x units produced and sold, the profit is P = (s − b)x − a. The break-even point (P = 0) occurs at x = a/(s − b) units. Below this quantity the firm suffers a loss; above it, profit grows. The rate of change dP/dx = s − b, so profit grows faster when selling price exceeds variable cost by a larger margin.

A tank starts with 250 kg of salt. Brine with 200 g/litre flows in at 25 litres/min (adding 5 kg/min) while mixed brine flows out at 25 litres/min (removing y/40 kg/min). This gives dy/dt + y/40 = 5, a linear ODE. Solving with initial condition y(0) = 250 yields y = 200 + 50e^(−t/40), showing the salt content decreases from 250 kg but never falls below 200 kg.

The appendix notes that many situations remain unmodelled because they are too complex or lead to mathematically intractable models. There are no good guidelines for choosing or estimating parameters/variables. Large or complex systems (environment, oceanography, pollution control) pose special problems. Powerful computers have expanded the scope of modelling, but fitting any data can require five or six arbitrarily chosen parameters, so minimising the number of parameters is important.

The appendix lists nine motivating problems: finding the width of a river without crossing it; finding the optimal angle for a shot-put considering thrower height, air resistance, and gravity; finding the height of an inaccessible tower; estimating the surface temperature of the Sun; explaining why heart patients should not use a lift; finding the mass of the Earth; estimating India's pulse yield from standing crops; estimating blood volume inside the human body; and estimating India's population for a future year.

The appendix uses trigonometry (angle of elevation/depression) in Example 1; matrix multiplication in Examples 2 and 3; elementary row operations on a matrix system in Example 4; linear programming with feasible-region corner-point evaluation in Example 5; differentiation and break-even analysis in Example 6; and solving a first-order linear differential equation in Example 7.

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