Mathematics: Class 11 — Chapter 16: Mathematical Modelling

A mathematical model is a simplified representation of a real-world situation using mathematical symbols, equations, or geometric structures. It allows us to analyze, understand, and predict the behavior of complex systems.

• Real-world Problem: The initial situation or question identified in the physical world.
• Mathematical Formulation: Translating the real-world parameters into mathematical variables, equations, or inequalities.
• Mathematical Solution: Solving the formulated mathematical equations.
• Interpretation & Validation: Translating the mathematical solution back into the real-world context and testing its accuracy against observed data.

Mathematical Problem Statement: The angle of elevation of the top of a tower from a point $O$ on the ground, which is $450\text{ m}$ away from the foot of the tower, is $40^\circ$. Find the height of the tower.

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Step-by-Step Algorithmic Solution Trace:

1. Step 1 (Understand the Problem): Let $h$ denote the height of the tower, $d$ denote the horizontal distance from point $O$ to the foot of the tower ($d = 450\text{ m}$), and $\theta$ denote the angle of elevation ($\theta = 40^\circ$).
2. Step 2 (Formulation): Express the relationship geometrically using a right-angled triangle. The trigonometric ratio connecting height, distance, and angle of elevation is:$$\tan \theta = \frac{h}{d} \implies h = d \tan \theta$$
3. Step 3 (Solution): Substitute the known values into the equation:$$h = 450 \times \tan 40^\circ \approx 450 \times 0.839 = 377.6\text{ m}$$
4. Step 4 (Interpretation): The height of the tower is approximately $378\text{ m}$.

Final Evaluation Output: Height of the tower $\approx 378\text{ m}$

In 1875, an extra bridge was built joining land areas A and B, changing the degree of these vertices to even. This left only two odd vertices (C and D), making it possible to traverse the town crossing each bridge exactly once.

1. Understanding the Problem: Clearly defining the real-world system, its boundaries, and the specific question to be answered.
2. Formulation:
   • Identifying Relevant Factors: Distinguishing between essential parameters and non-essential variables.
   • Mathematical Description: Establishing equations, inequalities, or geometric relations between the chosen parameters.
3. Finding the Solution: Solving the mathematical equations using algebraic, calculus, or graphical methods.
4. Interpretation and Validation: Translating the mathematical solution back to the physical context and comparing it with experimental data to measure the error.

We model the motion of a simple pendulum consisting of a mass (bob) attached to a string of length $l$ under gravity $g$.

• Formulation:
  • Through experimentation, we observe that the mass $m$ of the bob does not affect the period of oscillation $T$, whereas the length $l$ does.
  • Plotting experimental values of $T$ against $l$ yields a parabolic curve, suggesting the relation:$$T^2 = kl$$
  • The constant of proportionality is found to be $k = \frac{4\pi^2}{g}$, leading to the classical formula:$$T = 2\pi\sqrt{\frac{l}{g}}$$
• Validation:

  We compare calculated values with experimental measurements:

  Length ($l$)	Calculated Period ($T$)	Experimental Period ($T$) [Mass = 385g]	Absolute Error
  $225\text{ cm}$	$3.04\text{ s}$	$3.056\text{ s}$	$0.016\text{ s}$
  $275\text{ cm}$	$3.36\text{ s}$	$3.371\text{ s}$	$0.011\text{ s}$

  The small error validates the model. The minor discrepancy arises from neglecting air resistance and the mass of the string.

Mathematical Problem Statement: A farmhouse uses at least $800\text{ kg}$ of special food daily, which is a mixture of corn and soyabean. The compositions and costs are given below:

Material	Protein per Kg	Fibre per Kg	Cost per Kg
Corn	$0.09$	$0.02$	Rs 10
Soyabean	$0.60$	$0.06$	Rs 20

The dietary requirements stipulate at least 30% protein and at most 5% fibre. Determine the daily minimum cost of the food mix.

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Step-by-Step Algorithmic Solution Trace:

1. Step 1 (Identify Variables): Let $x$ be the amount of corn (in kg), $y$ be the amount of soyabean (in kg), and $z$ be the total cost.
2. Step 2 (Formulate Objective Function & Constraints):
   • Objective Function: Minimize $z = 10x + 20y$
   • Total food constraint: $x + y \ge 800$
   • Protein constraint (at least 30%):$$0.09x + 0.60y \ge 0.30(x + y) \implies 0.21x - 0.30y \le 0$$
   • Fibre constraint (at most 5%):$$0.02x + 0.06y \le 0.05(x + y) \implies 0.03x - 0.01y \ge 0$$
3. Step 3 (Solve Graphically): Plotting the inequalities on a coordinate plane, the feasible region is bounded, and the minimum value of $z$ occurs at the intersection point of the boundary lines:$$x = 470.6\text{ kg},\quad y = 329.4\text{ kg}$$$$z = 10(470.6) + 20(329.4) = 11294$$
4. Step 4 (Interpretation): The minimum daily cost is Rs 11,294, achieved by mixing $470.6\text{ kg}$ of corn and $329.4\text{ kg}$ of soyabean.

Final Evaluation Output: Minimum Cost = Rs 11,294 at $(470.6, 329.4)$

Mathematical Problem Statement: Model the population of a country over time and predict the population after 10 years, given an initial population of $250,000,000$, a constant birth rate $b = 0.02$, and a constant death rate $d = 0.01$.

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Step-by-Step Algorithmic Solution Trace:

1. Step 1 (Formulation): Let $P(t)$ be the population at year $t$. The population at year $t+1$ is given by:$$P(t+1) = P(t) + B(t) - D(t)$$where $B(t)$ is the number of births and $D(t)$ is the number of deaths.
2. Step 2 (Introduce Rates): Define birth rate $b = \frac{B(t)}{P(t)}$ and death rate $d = \frac{D(t)}{P(t)}$. Substituting these:$$P(t+1) = P(t) + bP(t) - dP(t) = (1 + b - d)P(t)$$
3. Step 3 (Derive General Solution): By induction:$$P(1) = (1 + b - d)P(0)$$$$P(2) = (1 + b - d)^2 P(0)$$$$P(t) = (1 + b - d)^t P(0) = P(0)r^t$$where $r = 1 + b - d$ is the growth rate (or Malthusian parameter).
4. Step 4 (Calculate Solution): For $P(0) = 250,000,000$, $b = 0.02$, $d = 0.01 \implies r = 1.01$:$$P(10) = 250,000,000 \times (1.01)^{10} \approx 276,155,531.25$$
5. Step 5 (Interpretation & Validation): Since a fractional person ($0.25$) is physically impossible, we approximate the population to $276,155,531$. The model assumes constant rates and no migration, which are simplifications of real-world dynamics.

Final Evaluation Output: Predicted Population $\approx 276,155,531$