Mathematical Modelling (Appendix)

A surveyor cannot climb a tower with a measuring tape, a meteorologist cannot bottle tomorrow's weather, and a third umpire cannot rewind a cricket ball's flight and watch it again in slow motion through a batsman's leg. In every one of these situations the real problem is too big, too distant, or too fast to touch directly — so we build a smaller, mathematical stand-in for it, work out that stand-in's answer, and translate the answer back into the language of the real world. This translation process is Mathematical Modelling, and this appendix chapter of NCERT Class 11 is where the idea is made explicit for the first time: not as a single formula but as a discipline, a repeatable four-step cycle that mathematicians, physiologists, agronomists and cricket umpires all use whether or not they call it by this name. This chapter is deliberately different from the rest of your Class 11 syllabus: there is no new formula to memorise for the board exam. Instead you will watch the same four-step cycle — understanding the problem, formulation, finding the solution, and interpretation/validation — applied to four genuinely different situations: measuring a tower with trigonometry, Euler's Königsberg bridge puzzle that gave birth to graph theory, timing a simple pendulum with $T = 2\pi\sqrt{l/g}$ , minimising cattle-feed cost with linear inequalities, and projecting a population forward with an exponential growth law. We will follow the NCERT text faithfully, then go one notch further: prove why the pendulum's period formula takes the shape it does dimensionally, formalise Euler's odd-vertex theorem as a rigorous argument, and practise the modelling cycle on fresh problems so "formulate, solve, validate" becomes a habit. Framed honestly, none of this is examined the way calculus or coordinate geometry is — but it is exactly the thinking that separates a student who can only apply formulas from one who can build a model when no formula yet exists.

Key takeaways

Mathematical modelling is the process of translating a real-life problem into mathematical form, solving the mathematical version, and translating the answer back into the real-world context.
The modelling cycle has four steps: Understanding the problem (identify what is asked), Formulation (identify relevant factors and write an equation/inequality/figure connecting them), Finding the solution (solve the mathematical formulation), and Interpretation/Validation (compare with real data; accept, reject or improve the model).
Example 1 (the tower problem) uses $\tan\theta = h/d$ , so $h = d\tan\theta$ ; with $d = 450$ m and $\theta = 40°$ , the height comes out to $h \approx 377.6$ m, interpreted as approximately $378$ m.
Example 2 (the Königsberg bridge problem) shows Leonhard Euler in 1736 modelling a city of land masses and seven bridges as a network of vertices and arcs; a vertex is odd if an odd number of arcs meet there, and a single walk crossing every bridge exactly once is possible only if there are $0$ or $2$ odd vertices — Königsberg had $4$ , so no such walk existed.
Example 3 (simple pendulum) finds, by experiment, that the period $T$ does not depend appreciably on the bob's mass but does depend on the string's effective length $l$ , leading to the model $T^2 = kl$ and, with $k = 4\pi^2/g$ , the formula $T = 2\pi\sqrt{l/g}$ .
Example 4 (cattle-feed cost) models a real constrained-optimisation problem as a linear programming problem: minimise $z = 10x + 20y$ subject to $x + y \geq 800$ and nutrient constraints, solved graphically to give a minimum cost of Rs $11294$ at $x = 470.6$ kg corn, $y = 329.4$ kg soyabean.
Example 5 (population projection) models population growth as $P(t) = P(0)\,r^{t}$ , an exponential function, where $r = 1 + b - d$ is the Malthusian growth rate built from the birth rate $b$ and death rate $d$ .
Validation compares the model's predicted values against measured/observed values; the gap between them is the error, and a model is accepted only if that error is acceptably small — otherwise the assumptions are revisited and a new model is built.
Because every model rests on simplifying assumptions (a massless string, no migration, a constant growth rate), no model is ever "the" complete truth — it is a good-enough approximation for the questions it was built to answer.
Key idea: mathematical modelling turns a real problem into a solvable mathematical one by cycling through understand, formulate, solve and validate/interpret, accepting or refining the model based on how well its predictions match reality.

The core insight

Mathematical modelling is the disciplined four-step cycle — understand the real problem, formulate it as an equation, inequality or geometric structure, solve the resulting mathematical problem, and interpret/validate the solution against real data — repeated and refined until the model's predictions agree with observation closely enough to be useful.

What this chapter covers

The chapter opens with A.2.1 Introduction, motivating why translating real-world problems into mathematics has become indispensable as computing power has grown, walking through the tower-height problem as a first, informal illustration before naming the steps.

A.2.2 Preliminaries widens the lens: modelling is not new (Egyptian, Babylonian, Greek and Indian geometry all modelled the physical world), and it lists domains where mathematics studies a situation — blood flow, LBW decisions in cricket, weather prediction, crop-yield estimation. It works the tower example in full four-step detail, then presents the celebrated Königsberg bridge problem, where Euler's network model launches graph theory.

A.2.3 What is Mathematical Modelling? gives the formal definition and demonstrates the full cycle — Understanding, Formulation, Finding the solution, Interpretation/Validation — on the simple pendulum ( $T = 2\pi\sqrt{l/g}$ ), a linear programming cattle-feed problem, and an exponential population growth model. It closes with the flowchart: Assumptions/Axioms to Formulation to Solution to Interpretation to Validation, a "Satisfied?" decision looping back to Formulation on No, or exiting to Stop on Yes. Our "Core ideas" section reorganises this material around the cycle itself, since that cycle — not any single formula — is the actual content taught. "Derivations" proves two central results rigorously: the pendulum formula by dimensional reasoning, and Euler's odd-vertex theorem. The worked examples then let you run the cycle on fresh numbers.

The core ideas

What a mathematical model is

NCERT's definition is precise: a mathematical model is a representation which comprehends a situation, and mathematical modelling is an attempt to study some part (or form) of a real-life problem in mathematical terms. Conversion of a physical situation into mathematics, with suitable simplifying conditions, is mathematical modelling. NCERT calls it "a technique and the pedagogy taken from fine arts and not from the basic sciences" — a craft of judgement, not a fixed algorithm. Two people modelling the same situation may build two different, both-useful models, depending on which factors they keep and discard. A model is always built by throwing information away. The real world is "far too complex to understand and describe completely," so a modeller picks one or two main factors believed to drive the behaviour of interest and ignores the rest, expecting the simplified model to still capture the essential characteristic of the problem. This is not a weakness of modelling — it is the point: a model with every conceivable factor included would be as hard to solve as the original problem.

The four-step modelling cycle

NCERT's four steps, illustrated throughout the chapter on the pendulum, tower, bridge, feed-cost and population examples, are:

1. Understanding the problem. Study the real-life situation and identify what quantity is being asked for and which physical quantities are involved. For the pendulum this step produces the precise statement: "How do we find the period of oscillation of the simple pendulum?"

2. Formulation, in two parts. First, identifying the relevant factors — deciding, often by experiment, which quantities matter (equal-length strings with different bob masses show the pendulum's period barely changes with mass, so mass is dropped; equal masses with different lengths show a clear dependence, so length is kept). Second, mathematical description — writing an equation, inequality, or figure connecting the retained quantities, e.g. $\tan\theta = h/d$ for the tower or $T^2 = kl$ for the pendulum.

3. Finding the solution. The formulation rarely hands you the answer directly; you must solve the equation, evaluate an inequality's feasible region, or apply a theorem — arithmetic for the tower, a graphical linear-programming solution for the feed-cost problem, substitution for the pendulum.

4. Interpretation/Validation. Translate the answer back into the real problem's language, and, where data exists, compare predicted against measured values. The gap is the error; NCERT computes it for the pendulum ( $3.371 - 3.36 = 0.011$ s, small enough to accept). If unacceptable, the modeller revisits the assumptions and repeats the cycle — why NCERT draws it as a loop: Assumptions to Formulation to Solution to Interpretation to Validation to a Yes/No branch, looping back to Formulation on "No," stopping only on "Yes."

Figure

A flowchart of the mathematical modelling cycle showing Start leading to Assumptions and Axioms leading to Formulation leading to Solution leading to Interpretation leading to Validation leading to a decision Satisfied which loops back to Formulation on No and proceeds to Stop on Yes

The tower problem — modelling with trigonometry

A surveyor wants the height $h$ of a tower without climbing it. He can measure the horizontal distance $d$ from a point $O$ to the foot of the tower, and the angle of elevation $\theta$ of the top as seen from $O$ . If $AB$ is the tower and $OA = d$ , right-triangle trigonometry gives

\tan\theta = \frac{h}{d} \quad\Rightarrow\quad h = d\tan\theta. \qquad (1)

With $d = 450$ m and $\theta = 40°$ , $h = 450\tan 40° = 450 \times 0.839 = 377.6$ m, interpreted as approximately $378$ m. This maps cleanly onto the four-step cycle: Step 1 identifies $h$ , $d$ , $\theta$ ; Step 2 writes equation (1); Step 3 substitutes numbers; Step 4 rounds sensibly and states the answer in the tower's own terms.

The Königsberg bridge problem — modelling with a network

Eighteenth-century Königsberg straddled the Pregel river, its two islands connected to the two banks and to each other by seven bridges. Citizens wondered whether one could walk through the city crossing every bridge exactly once. Euler's insight in 1736 was to strip away every irrelevant detail — street names, distances, shapes — and keep only connectivity: each land mass (two banks, two islands) becomes a vertex (a dot), each bridge an arc (a line joining two vertices). This abstraction is itself the modelling step: Königsberg's geography becomes a network of $4$ vertices and $7$ arcs. Call the vertices $A$ , $B$ (banks) and $C$ , $D$ (islands). Counting bridges: $3$ arcs meet $A$ , $3$ meet $B$ , $5$ meet $C$ , $3$ meet $D$ — every vertex has an odd number of arcs. Euler proved a walk crossing every arc exactly once (an "Eulerian walk") is possible only if the network has zero or two odd vertices; Königsberg's four odd vertices meant no such walk existed — proved impossible, not merely undiscovered. When an eighth bridge joined $A$ and $B$ directly, both became even, leaving only $C$ and $D$ odd — exactly two — so the walk became possible. This one puzzle seeded graph theory, now used for railway networks, circuit design, and internet routing.

Figure

Euler network of Konigsberg with four vertices A B C and D and seven arcs, showing the number of bridges meeting each vertex and marking every vertex as odd since an odd number of arcs meets each one

The simple pendulum — modelling with experiment and a power law

A simple pendulum is a bob on a string of effective length $l$ swinging under gravity $g$ ; NCERT asks for its period of oscillation $T$ . Step 1 names the candidate factors: $T$ , mass $m$ , length $l$ , and $g$ . Step 2 begins experimentally: holding $l$ fixed and varying $m$ shows no appreciable change in $T$ , so mass is discarded; holding $m$ fixed and varying $l$ shows a clear dependence, so $l$ is retained along with $g$ (constant at a given place). Plotting $T$ against $l$ gives a parabola-shaped curve, suggesting $T^2 = kl$ ; experiment (and, as derived below, dimensional reasoning) fixes $k = 4\pi^2/g$ , giving

T = 2\pi\sqrt{\frac{l}{g}}. \qquad (2)

Step 4 is where this example earns its place: NCERT computes $T$ from equation (2) for $l = 225$ cm and $l = 275$ cm, obtaining $T = 3.04$ s and $T = 3.36$ s, then compares against measured periods from real pendulums of different bob masses ( $3.042$ , $3.056$ , $3.352$ , $3.371$ s). The largest discrepancy, for $l = 275$ cm, $m = 385$ g, is $3.371 - 3.36 = 0.011$ s — small enough to accept the model, the residual attributed to the string's own mass and air resistance.

Linear programming and exponential growth as models

Not every model is a single equation. NCERT's cattle-feed problem models cost-minimisation as a linear programming problem: minimise $z = 10x + 20y$ (cost in rupees, $x$ = kg corn, $y$ = kg soyabean) subject to $x + y \geq 800$ (minimum daily food) plus a protein and a fibre inequality — solved graphically at a corner point of the feasible region. The population problem instead models growth as an exponential function $P(t) = P(0)r^{t}$ , $r = 1 + b - d$ combining birth rate $b$ and death rate $d$ . Both make the same point with different machinery: identify the factors, write the relationship precisely, solve it, and interpret the raw number sensibly — rounding $276{,}155{,}531.25$ people down to $276{,}155{,}531$ , since " $0.25$ of a person" is meaningless.

Derivations

Derivation ADeriving the pendulum period formula T equals 2 pi root l over g

NCERT states $T = 2\pi\sqrt{l/g}$ as an experimentally-fitted law, obtained by plotting measured periods against string length, recognising a parabola-like curve consistent with $T^2 = kl$ , then quoting $k = 4\pi^2/g$ . Here we justify why $k$ takes that form, combining the finding that mass drops out and length matters with a dimensional argument, so the formula is understood, not just fitted. Once you accept, from experiment, that $T$ depends only on $l$ and $g$ , dimensional consistency pins down the shape of the relationship up to one undetermined constant, and a short physical argument fixes that constant to $2\pi$ .

Step 1. Restate the experimental finding as a modelling assumption: $T$ depends only on the effective length $l$ (dimension $[L]$ ) and the acceleration due to gravity $g$ (dimension $[LT^{-2}]$ ); mass has been shown not to matter and is dropped. This is NCERT's "identifying the relevant factors" sub-step, made explicit.

Step 2. Propose the general power-law form suggested by the smooth T-versus-l curve: $T = k\, l^{a} g^{b}$ for constants $k$ (dimensionless), $a$ , $b$ to be determined. This is the "mathematical description" sub-step — choosing a functional form connecting the retained quantities.

Step 3. Match dimensions on both sides. The left side $T$ has dimension $[T^1]$ . The right side has dimension $[L]^{a}\,[LT^{-2}]^{b} = [L^{a+b}T^{-2b}]$ . Equating powers of $L$ and $T$ separately: $a + b = 0$ , and $-2b = 1$ , so $b = -\tfrac12$ and $a = \tfrac12$ .

Step 4. Substitute back: $T = k\, l^{1/2} g^{-1/2} = k\sqrt{l/g}$ . Dimensional analysis alone cannot determine the pure number $k$ — it fixes only how $T$ scales with $l$ and $g$ , already reproducing NCERT's fitted relation $T^2 = kl$ up to the value of $k$ .

Step 5. Fix $k$ using the geometry of small oscillations. For small swing angles the bob's motion along the arc approximates simple harmonic motion with restoring acceleration $g\sin\theta \approx g\theta$ , the same functional form as uniform circular motion projected onto a diameter, of angular frequency $\omega = \sqrt{g/l}$ . Since $T = 2\pi/\omega$ , this gives $T = 2\pi\sqrt{l/g}$ , so $k = 2\pi$ exactly (the full geometric argument belongs to simple harmonic motion, beyond this chapter, but the dimensional skeleton above is squarely within it).

Step 6 (conclusion). Combining the dimensional result $T = k\sqrt{l/g}$ with $k = 2\pi$ recovers exactly NCERT's Equation (2): $T = 2\pi\sqrt{l/g}$ . This is a second, complementary route to a model — where experiment alone can only fit a curve, dimensional reasoning plus a physical argument derives its exact form, and the two agreeing is itself a form of validation.

Go deeper · Derivation B · 2 minEulers odd vertex theorem for one trip crossing every bridge exactly once

Derivation BEulers odd vertex theorem for one trip crossing every bridge exactly once

NCERT asserts, without a full proof, that a network can be traversed by a single walk crossing every arc exactly once if and only if it has zero or two odd vertices (a vertex is odd if an odd number of arcs meet there), and that Königsberg's four odd vertices rule out such a walk. Here we prove the "only if" direction rigorously — that more than two odd vertices makes such a walk impossible — since that is exactly what NCERT's argument needs, and its logic is genuinely non-obvious.

Step 1. Set up the counting argument. Suppose a walk exists that starts at some vertex, ends at some vertex (possibly the same one), and crosses every arc of the network exactly once. Consider any vertex $V$ that is neither the start nor the end of this walk.

Step 2. Every time the walk passes through $V$ , it uses exactly two arcs there: one to arrive, one different arc to leave, since no arc may be reused. Each such pass-through consumes arcs at $V$ in pairs.

Step 3. Since the walk crosses every arc exactly once, and every arc at $V$ must be used during some visit (arrival or departure), the arcs at $V$ are used up entirely by these arrival-departure pairs. Hence the total number of arcs meeting $V$ must be even — $V$ must be an even vertex.

Step 4. Contrapositive: if $V$ is odd, it cannot be a pass-through vertex — it must be the start or the end. Since a single walk has exactly one start and one end, at most two vertices of the network can be odd for such a walk to exist.

Step 5. Apply this to Königsberg. The network has four vertices $A$ , $B$ , $C$ , $D$ with $3$ , $3$ , $5$ , $3$ arcs respectively — all four odd. By Step 4, a walk using every arc could have at most two odd vertices acting as start and end; since all four are odd, this is impossible.

Step 6 (conclusion, and the converse). This proves NCERT's claim directly: more than two odd vertices makes an Eulerian walk impossible, regardless of routing. Adding an eighth bridge joining $A$ and $B$ reduces the odd-vertex count to two, exactly the boundary the theorem allows — consistent with a qualifying walk existing afterward (the converse, that zero or two odd vertices guarantees such a walk in a connected network, is the deeper half of Euler's theorem, assumed here rather than proved).

Worked examples

Board level - Modelling a ramp's angle with the four-step cycle

A wheelchair ramp is to be built so its top rests on a platform $1.2$ m above the ground, and the foot of the ramp is $6$ m horizontally from directly below the platform edge. Using the modelling cycle, find the angle of inclination $\theta$ the ramp makes with the ground.

Step 1. Understanding the problem. What angle does the ramp make with the horizontal ground? Known: vertical rise ( $1.2$ m) and horizontal run ( $6$ m); $\theta$ is unknown.

Step 2. Formulation. Relevant factors: $h = 1.2$ m, $d = 6$ m, angle $\theta$ . Picturing the ramp, platform edge, and ground as a right triangle, the mathematical description is $\tan\theta = h/d$ .

Step 3. Finding the solution. Substitute: $\tan\theta = 1.2/6 = 0.2$ . Then $\theta = \tan^{-1}(0.2) \approx 11.3°$ .

Step 4. Interpretation/Validation. The ramp rises at approximately $11.3°$ from the horizontal, exceeding common accessibility guidelines (which cap ramp angles near $4.8°$ for a $1{:}12$ rise-to-run ratio), so a real designer would validate this output against the code and lengthen the run rather than accept the steeper angle — precisely the Interpretation/Validation branch of the cycle. The ramp's angle of inclination is approximately $\boxed{11.3°}$ .

Board level - Modelling population growth over two years

A town has population $50{,}000$ , birth rate $b = 0.03$ and death rate $d = 0.01$ per year, with no migration. Using NCERT's exponential population model, find the population after $2$ years.

Step 1. Understanding: find $P(2)$ , given $P(0) = 50{,}000$ , $b = 0.03$ , $d = 0.01$ .

Step 2. Formulation. $P(t) = P(0)\,r^{t}$ where $r = 1 + b - d = 1 + 0.03 - 0.01 = 1.02$ .

Step 3. Finding the solution. $P(2) = 50{,}000 \times (1.02)^2 = 50{,}000 \times 1.0404 = 52{,}020$ .

Step 4. Interpretation/Validation. Population must be a whole number; $52{,}020$ is already an integer, so no rounding issue arises (unlike NCERT's own $276{,}155{,}531.25$ case). The model predicts growth of about $2{,}020$ people in two years under these assumptions. The population after 2 years is approximately $\boxed{52{,}020}$ .

Higher-order thinking (HOTS) - Testing and repairing a linear model with real data

A shopkeeper models weekly ice-cream sales $S$ as a linear function of the day's maximum temperature $x$ (Celsius): $S = 8x - 120$ , fitted from $(30°\text{C}, 120)$ and $(35°\text{C}, 160)$ . She then records: at $28°C$ , $95$ units sold; at $32°C$ , $138$ units; at $20°C$ , $10$ units. (a) Verify the model reproduces its two fitting points. (b) Validate against the three new points, computing each error. (c) Using the $20°C$ prediction, explain why a purely linear model is a poor global choice here, and what assumption is failing.

Step 1. Understanding the problem. We are given $S = 8x - 120$ built from two points, and asked to validate it against three fresh observations, then diagnose a structural weakness — the Interpretation/Validation and "loop back to Formulation" stages of the cycle.

Step 2. Verify against the fitting points. At $x = 30$ : $S = 240 - 120 = 120$ , matching. At $x = 35$ : $S = 280 - 120 = 160$ , matching. The model reproduces both fitting points exactly, since it was constructed from them.

Step 3. Validate against new data. At $x = 28$ : predicted $S = 104$ ; observed $95$ ; error $= 9$ units ( $9.5\%$ ). At $x = 32$ : predicted $S = 136$ ; observed $138$ ; error $= -2$ units ( $1.4\%$ ), small and acceptable. At $x = 20$ : predicted $S = 40$ ; observed $10$ ; error $= 30$ units, $300\%$ of observed — a wildly poor prediction.

Step 4. Diagnose the structural failure. Near $30\text{–}32°C$ the errors are small, so locally the model is reasonably validated. But at $20°C$ it badly overshoots, and if $x$ dropped toward $15°C$ the formula would predict $S = 0$ , and below that, negative sales — a physical impossibility. The hidden assumption that broke: the model treats demand as falling in a straight line forever, but real demand flattens toward a small floor value rather than crossing zero. It should be re-formulated with a curve that flattens for low temperatures, e.g. $S = c + k(x - x_0)^2$ above some threshold $x_0$ , not one straight line extended indefinitely. The three errors are $9$ , $-2$ , and $30$ units; the model fails at low temperatures because it predicts negative sales, so it must be re-formulated, not merely re-fitted. Timing note: allow 10-12 minutes — this problem has three separate arithmetic checks plus a conceptual diagnosis, which is the slowest part.

Common traps

Treating the model as automatically "correct" because the algebra is correct — it can be arithmetically flawless and still be a bad model of reality if the wrong factors were chosen at Formulation (students often keep mass "to be safe" even after experiment shows it doesn't matter).
Skipping Interpretation/Validation entirely and stopping at a numeric answer — NCERT's population example shows why this fails: $P(10) = 276{,}155{,}531.25$ must be interpreted before it means anything.
Confusing "the model fits the data used to build it" with "the model is validated" — a model built from two points always passes through those points; real validation needs independent data it was not fitted to.
Assuming Euler's odd-vertex rule needs the network connected only "sort of" — the walk must traverse the whole network in one trip, so a network split into pieces still cannot have a single all-arcs walk.
Believing every real situation needs an equation — the cattle-feed example needed a system of inequalities solved graphically, showing "mathematical form" can mean equations, inequalities, figures or networks.

Common trapThe single biggest misconception is thinking a mathematical model is meant to be exactly true rather than usefully close

Every model here rests on assumptions known, in advance, to be only approximately true: the pendulum ignores string mass and air resistance; the population model assumes constant birth/death rates and zero migration; the tower model assumes level ground and a vertical tower. NCERT is explicit that the resulting small errors (the pendulum's $0.011$ -second gap) are expected, not failures — a model is accepted because its error is small enough for the purpose, not because it is exactly correct. The goal is never a perfect mirror of reality but the simplest structure whose predictions are close enough to be useful, and "close enough" is a judgement call the modeller must justify.

You should be able to…

State the four steps of the modelling cycle in order and explain what happens at each.
Translate a real-world measurement problem into a trigonometric equation and solve it, as in the tower example.
Explain why Euler's network abstraction of Königsberg's bridges made a geographic puzzle into a solvable counting problem.
State and apply the odd-vertex condition for a single walk crossing every arc exactly once.
Describe how $T = 2\pi\sqrt{l/g}$ was arrived at by combining experiment (dropping mass, keeping length) with a fitted power law.
Compute the error between a predicted and measured value, and explain what validating or rejecting a model means.
Set up a linear programming formulation (objective plus constraint inequalities) from a word problem, as in the cattle-feed example.
Apply $P(t) = P(0)r^t$ with $r = 1 + b - d$ to project a population forward, and interpret non-integer outputs sensibly.
Recognise, given a fitted model and new data, when it should be accepted, refined, or replaced.

Ask the tutor: Want to pick a real situation from your own life — a daily commute, a phone's battery drain, a cricket score chase — and walk through all four modelling steps on it together?

Is mathematical modelling actually tested in the Class 11 board exam?

Not usually as a standalone numeric problem the way trigonometry or calculus questions are. This chapter is an NCERT appendix, meant to build the habit of translating real situations into mathematics rather than hand you one more formula to memorise. That said, the underlying ideas — using $\tan\theta = h/d$ for heights and distances, recognising exponential growth $P(t) = P(0)r^t$ , or setting up linear programming inequalities — do appear as scored problems elsewhere in your syllabus (Trigonometric Functions, Linear Inequalities, Sequences and Series). Treat this chapter as the "why" behind those techniques rather than a separate exam topic.

Why did NCERT use the Konigsberg bridge problem instead of a more modern example of graph theory?

Because it is the cleanest illustration of what modelling accomplishes: an unsolved practical puzzle (cross every bridge exactly once?) becomes solvable the instant it is translated into an abstract structure — a network of dots and lines — that strips away irrelevant detail (street shapes, bridge lengths) and keeps only what matters (which land masses connect, and how many bridges join each pair). Euler's 1736 solution is also historically the first paper in graph theory, so the example does double duty: it teaches modelling and introduces a whole branch of mathematics used today in GPS routing, social networks, and circuit design.

If a model like the pendulum formula has an error, why do we still use it instead of finding the exact answer?

Because there usually is no simpler "exact" alternative that stays usable. The true motion of a real pendulum depends on the string's own mass, its stretchiness, air resistance, and tiny variations in local gravity — accounting for all of that produces equations far too complicated to solve by hand. NCERT's validation showed the simplified formula $T = 2\pi\sqrt{l/g}$ is off by roughly a hundredth of a second against real measurements, negligible for almost every practical purpose. Modelling is precisely about accepting a known, small, controlled error in exchange for a solvable structure — provided you report that trade-off honestly, which is what Validation is for.

Whats the difference between formulation and finding the solution in the modelling cycle - arent they the same step?

No, and mixing them up is a common confusion. Formulation translates the real problem into mathematics — deciding which quantities matter and writing the equation, inequality or network connecting them (writing $\tan\theta = h/d$ , or the vertex-and-arc network for Königsberg). Finding the solution is a purely mathematical exercise performed after translation is complete — solving that equation for $h$ , or checking the odd-vertex condition. You could hand the formulation to someone with no knowledge of the original problem, and they could still carry out the solution using only mathematics; interpretation is what reconnects the pure-math answer back to towers, bridges, or pendulums.