Chapter 16 — Appendix A2: Mathematical Modelling

The appendix defines a mathematical model as a mathematical description of some real-life situation. Mathematical modelling is the process of creating such a model, solving it, and using it to analyse and solve the original real-world problem.

Step 1 is understanding the problem and making simplifying assumptions. Step 2 is mathematical description and formulation—defining variables, writing equations or inequalities, gathering data into tables, making graphs, or calculating probabilities. Step 3 is solving the mathematical problem using mathematical techniques. Step 4 is interpreting the solution in the real-life context. Step 5 is validating the model by checking whether the results make sense; if they do not, the assumptions are revised and the process repeats.

If the solution does not make sense in the real situation, the modeller reconsiders the assumptions made in Step 1 and revises them to be more realistic, possibly including factors that were ignored earlier, then repeats the modelling cycle.

Twenty fishes are caught, marked, and released back into the lake. A second sample of 50 fishes is drawn; 5 are found to be marked. Since 5/50 = 1/10 of the sample is marked, the same fraction of the whole population is assumed to be marked, giving a total population estimate of 20 × 10 = 200 fishes. A key assumption is that the marked fishes mix uniformly with the rest.

Rolling two dice produces 36 equally likely outcomes. Of all possible sums (2 through 12), the sum 7 can occur in 6 ways, giving it a probability of 6/36 = 1/6, which is higher than any other sum. The model therefore recommends guessing 7 repeatedly to maximise points.

Juhi pays ₹600 cash down and two monthly instalments of ₹610 each under the shopkeeper's scheme. Solving the simple-interest equation shows the shopkeeper charges approximately 13.14% per annum, which is higher than the bank's 10% per annum. Therefore, borrowing from the bank to pay cash is the more economical option.

The appendix lists: finding the width and depth of a river at an unreachable place, estimating the mass of the Earth and other planets, estimating the distance between Earth and any other planet, predicting the arrival of the monsoon, predicting stock market trends, estimating the volume of blood in the body, predicting a city's population after 10 years, estimating the number of leaves in a tree, estimating the ppm of different pollutants in a city's atmosphere, estimating the effect of pollutants on the environment, and estimating the temperature on the Sun's surface.

The appendix states that different assumptions used for simplifying a problem lead to different models. Modellers must balance simplification (for ease of solution) against accuracy. Because assumptions vary and simplifications inevitably omit some real-world factors, there are no perfect models—only better or worse ones.

The appendix explains that mathematicians and specialists in other fields share their knowledge and expertise to improve existing products, develop better ones, or predict the behaviour of certain products, making modelling inherently interdisciplinary.

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