Learn how to convert real-life problems into mathematics using matrices, calculus and linear programming — with visual models, step-marked solutions, MCQs, A-R, and PYQs.
Understanding mathematical modelling
In class XI, we have learnt about mathematical modelling as an attempt to study some part (or form) of some real-life problems in mathematical terms, i.e., the conversion of a physical situation into mathematics using some suitable conditions.
Mathematical modelling is an activity in which we make models to describe the behaviour of various phenomenal activities of our interest in many ways using words, drawings or sketches, computer programs, mathematical formulae etc.
In this chapter, we shall further study mathematical modelling of some real-life problems using techniques/results from matrix, calculus and linear programming.
Students are aware of the solution of word problems in arithmetic, algebra, trigonometry and linear programming etc. Sometimes we solve the problems without going into the physical insight of the situational problems. Situational problems need physical insight that is introduction of physical laws and some symbols to compare the mathematical results obtained with practical values.
Examples of problems that need mathematical modelling:
All of these problems can be solved and in fact have been solved with the help of Mathematics using mathematical modelling.
Some of the basic principles of mathematical modelling are listed below:
See mathematical modelling in action
The need for converting real problems into mathematics
Situational problems need physical insight — that is, introduction of physical laws and some symbols to compare the mathematical results obtained with practical values. To solve many problems faced by us, we need a technique — and this is what is known as mathematical modelling.
Mathematics is the language in which we can describe, analyse, and predict real-world phenomena.
| Problem | Modelling Approach |
|---|---|
| Width of a river | Trigonometry (angles, distances) |
| Optimal angle in shot-put | Calculus (maxima-minima) |
| Height of a tower | Trigonometry (angle of elevation) |
| Temperature of Sun's surface | Physics + Calculus |
| Mass of the Earth | Gravitation + Calculus |
| Yield of pulses | Statistics + Sampling |
| Volume of blood | Dilution method + Calculus |
| Population in 2020 | Exponential growth model |
Philosophical foundations and guidelines
The systematic procedure
Where mathematical modelling is used
The development of powerful computers and supercomputers has enabled us to mathematically model a large number of situations (even complex ones). Due to these fast and advanced computers, it has been possible to prepare more realistic models which can obtain better agreements with observations.
Key mathematical models used in real-life problems
Where A = purchase orders (clients × products), B = raw material per unit (products × raw materials)
Speed techniques for modelling problems
Ask: What is changing? What is fixed?
"Write relation, then differentiate"
x = a/(s − b)
A(m×n) × B(n×p) = C(m×p)
Evaluate Z at every vertex of feasible region
Rate of change = (inflow concentration × rate) − (outflow concentration × rate)
dy/dt + ky = Q → y(t) = Q/k + Ce^(−kt)
Avoid these in modelling problems:
Always interpret the result in the original context
Always verify:
Click "Show Answer" to reveal step-by-step solutions with marks
Q1. Find the height of a given tower using mathematical modelling.
Given physical situation is "to find the height of a given tower".
Let AB be the tower. Let PQ = h be the height of observer's eye and H be height of tower. Let α be angle of elevation and l be distance.
From right triangle: $\tan\alpha = \frac{H-h}{l}$
$H = h + l\tan\alpha$
Values of h, l, α are known to the observer.
If foot is not accessible, let β be angle of depression. Then $l = h\cot\beta$, giving $H = h + h\cot\beta\tan\alpha$.
Q2. A firm produces 3 products using 3 raw materials. Prepare a matrix model to determine raw material requirements.
Firm produces products P₁, P₂, P₃ using raw materials R₁, R₂, R₃ for clients F₁, F₂.
Let A = purchase orders matrix (2×3), B = raw material per unit matrix (3×3).
AB = matrix giving desired quantities of R₁, R₂, R₃ to fulfill orders of F₁, F₂.
Compare required raw material with available stock to determine feasibility.
Q3. Interpret the model in Example 2 in case A = [[6,4,6],[5,7,8]], B = [[3,4,0],[7,9,3],[5,12,7]] and available raw materials are 330, 455, 140 units.
AB = [[6·3+4·7+6·5, 6·4+4·9+6·12, 6·0+4·3+6·7], [5·3+7·7+8·5, 5·4+7·9+8·12, 5·0+7·3+8·7]]
AB = [[76, 132, 54], [104, 179, 77]]
Total R₁ = 76 + 104 = 180, R₂ = 132 + 179 = 311, R₃ = 54 + 77 = 131
Required (180, 311, 131) vs Available (330, 455, 140).
R₁ and R₂ are within limits, but R₃ is just within limit.
Q4. A firm has 330 units of R₁, 455 units of R₂ and 140 units of R₃. How many units of each product to make to utilize full raw material?
Let firm produce x units of P₁, y units of P₂ and z units of P₃.
$3x + 7y + 5z = 330$ (R₁)
$4x + 9y + 12z = 455$ (R₂)
$3y + 7z = 140$ (R₃)
Using row reduction: $x = 20$, $y = 35$, $z = 5$
Firm can produce 20 units of P₁, 35 units of P₂, and 5 units of P₃ to make full use of available raw material.
Q5. A manufacturer prepares medicines M₁ and M₂. Max 20000 of M₁, 40000 of M₂, 45000 total bottles. Time: 3 hrs per 1000 M₁, 1 hr per 1000 M₂, 66 hrs total. Profit: Rs 8/bottle M₁, Rs 7/bottle M₂. Maximise profit.
Let x = bottles of M₁, y = bottles of M₂.
Maximise $Z = 8x + 7y$
$x \leq 20000$, $y \leq 40000$, $x + y \leq 45000$
$3x + y \leq 66000$, $x \geq 0$, $y \geq 0$
Z(0,0)=0, Z(20000,0)=160000, Z(20000,6000)=202000
Z(10500,34500)=325500, Z(5000,40000)=320000, Z(0,40000)=280000
Q6. A company produces a new product with fixed cost a and variable cost b per unit. Selling price s per unit. Prepare a model to examine profitability.
x = units produced and sold, C = total cost, I = income, P = profit.
$C = a + bx$ (fixed + variable)
$I = sx$
$P = I - C = sx - (a + bx) = (s-b)x - a$
$\frac{dP}{dx} = s - b$ (rate of change of profit)
P = 0 → $x = \frac{a}{s-b}$ units
If s > b, profit increases with production.
Q7. A tank contains 1000 litres of brine with 250 kg salt. Brine with 200 g/l flows in at 25 l/min and mixture flows out at same rate. Find salt content at time t.
Let y(t) = amount of salt (kg) at time t (minutes). Initially y(0) = 250 kg.
Inflow rate = 25 × 200g = 5 kg/min
Outflow rate = 25 × (y/1000) = y/40 kg/min
$\frac{dy}{dt} = 5 - \frac{y}{40}$
$\frac{dy}{dt} + \frac{y}{40} = 5$ (linear differential equation)
Solution: $y = 200 + Ce^{-t/40}$
At t = 0, y = 250 → C = 50
$y(t) = 200 + 50e^{-t/40}$
As t → ∞, y → 200 kg. Salt content remains between 200 and 250 kg.
Select your answer and click "Check Answers"
Each question has an Assertion (A) and a Reason (R). Choose the correct option.
(A) Both A and R are true, and R is the correct explanation of A.
(B) Both A and R are true, but R is NOT the correct explanation of A.
(C) A is true, but R is false.
(D) A is false, but R is true.
Frequently asked questions in board exams
Q. Explain the five steps of mathematical modelling with an example.
Identify the physical situation (e.g., find height of tower).
Introduce variables, use laws (trigonometry): H = h + l tan α.
Substitute known values and find H.
Compare with actual measurement.
If good agreement, accept. Otherwise modify.
Q. A firm produces 3 products using 3 raw materials. Prepare a matrix model.
A = purchase orders (clients × products), B = raw material per unit (products × raw materials).
AB = matrix giving total raw materials needed.
Compare with available stock to plan production.
Q. A company has fixed cost Rs a and variable cost Rs b per unit. Selling price Rs s. Model the profitability and find break-even point.
C = a + bx
I = sx
P = I − C = (s−b)x − a
dP/dx = s − b
P = 0 → x = a/(s−b)
If x < a/(s−b), loss. If x > a/(s−b), profit.
Q. Solve the brine tank problem: dy/dt = 5 − y/40, y(0) = 250.
dy/dt + y/40 = 5
IF = e^(t/40)
y·e^(t/40) = 200e^(t/40) + C
y = 200 + Ce^(−t/40)
y(0) = 250 → C = 50
y(t) = 200 + 50e^(−t/40)
Q. A manufacturer makes medicines M₁ and M₂ with constraints. Maximise profit Z = 8x + 7y.
x ≤ 20000, y ≤ 40000, x+y ≤ 45000, 3x+y ≤ 66000
O(0,0), P(20000,0), Q(20000,6000), R(10500,34500), S(5000,40000), T(0,40000)
Z(R) = 8(10500) + 7(34500) = 325500 (max)
10500 units M₁, 34500 units M₂ for max profit Rs 3,25,500
Q. Find the height of a tower using angle of elevation and distance.
Let h = eye height, l = distance, α = angle of elevation, H = tower height.
tan α = (H−h)/l
H = h + l tan α
Let β = angle of depression. Then l = h cot β, so H = h + h cot β tan α.
Q. Interpret the matrix model AB = raw materials. If A = [[6,4,6],[5,7,8]], B = [[3,4,0],[7,9,3],[5,12,7]], compute AB.
(6·3+4·7+6·5, 6·4+4·9+6·12, 6·0+4·3+6·7) = (76, 132, 54)
(5·3+7·7+8·5, 5·4+7·9+8·12, 5·0+7·3+8·7) = (104, 179, 77)
Total R₁ = 180, R₂ = 311, R₃ = 131. Compare with available stock.
Q. State the principles of mathematical modelling.
Identify need, parameters, data, assumptions, governing principles.
Identify equations, calculations, solution.
Identify tests for consistency and utility.
Identify parameter values that can improve the model.