Question 1: What is Mathematical Modelling? Describe the main steps involved in the mathematical modelling process.

Solution:

Definition: Mathematical Modelling is the process of representing real-world problems in mathematical terms, solving them, and using the solution to understand, predict, or make decisions about the real-world situation.

Main Steps Involved:

1. Real-World Problem Formulation: Define the real-world context and determine what needs to be measured or predicted.
2. Simplifying Assumptions: Make reasonable assumptions to simplify the problem without losing essential features.
3. Mathematical Formulation: Convert words and assumptions into mathematical symbols, functions, and equations (e.g., linear equations, quadratic models).
4. Solving the Model: Apply algebraic, geometric, or arithmetic methods to obtain mathematical solutions.
5. Interpretation & Validation: Translate the solution back into the real-world context and test whether the results are practical and accurate. If not, refine the assumptions and repeat.

Question 2: A small business manufactures items where the fixed daily setup cost is Rs. 500 and the production cost per item is Rs. 20. (i) Formulate a mathematical model for the total daily cost C(x) for producing x items. (ii) Calculate the total cost for producing 45 items in a day. (iii) If the daily expenditure budget is Rs. 1700, how many items can be produced?

Solution:

Step 1: Mathematical Formulation

Let the number of items produced per day be x.

Fixed daily setup cost = Rs. 500

Variable cost for x items = 20 × x = 20x

Total daily cost function C(x) is given by:

C(x) = 500 + 20x

Step 2: Calculate cost for 45 items

Substitute x = 45 into the model equation:

C(45) = 500 + 20 × 45

C(45) = 500 + 900 = Rs. 1400

Step 3: Calculate items produced for a budget of Rs. 1700

Set C(x) = 1700:

1700 = 500 + 20x

20x = 1700 - 500

20x = 1200

x = 1200 / 20 = 60 items.

Conclusion: The mathematical model is C(x) = 500 + 20x. Producing 45 items costs Rs. 1400, and 60 items can be produced with a budget of Rs. 1700.

Question 3: A stone is dropped from a cliff of height 80 meters. Its height h(t) in meters after t seconds is modeled by the equation h(t) = 80 - 4.9t². (i) Find the height of the stone after 2 seconds. (ii) Calculate the time taken by the stone to hit the ground.

Solution:

Step 1: Given Model Equation

h(t) = 80 - 4.9t², where t is time in seconds and h(t) is height in meters.

Step 2: Find height at t = 2 seconds

Substitute t = 2 into the equation:

h(2) = 80 - 4.9 × (2)²

h(2) = 80 - 4.9 × 4

h(2) = 80 - 19.6 = 60.4 meters.

Step 3: Find time taken to hit the ground (h = 0)

Set h(t) = 0:

0 = 80 - 4.9t²

4.9t² = 80

t² = 80 / 4.9 = 800 / 49

t = √(800 / 49) = (20 × √2) / 7

Taking √2 ≈ 1.414:

t = (20 × 1.414) / 7 = 28.28 / 7 ≈ 4.04 seconds.

Conclusion: After 2 seconds, the height is 60.4 meters, and the stone hits the ground in approximately 4.04 seconds.

Question 4: A deposit of Rs. 10,000 is made at an interest rate of 10% per annum. (i) Write the mathematical models for total amount after 't' years under Simple Interest and Compound Interest (compounded annually). (ii) Compare the total amount after 3 years for both models.

Solution:

Step 1: Formulate the Models

Given Principal (P) = Rs. 10,000, Rate (r) = 10% per annum, Time = t years.

• Simple Interest Model (A_SI):
  Interest = (P × r × t) / 100 = (10000 × 10 × t) / 100 = 1000t
  A_SI(t) = P + Interest = 10000 + 1000t
• Compound Interest Model (A_CI):
  A_CI(t) = P × (1 + r/100)^t = 10000 × (1 + 10/100)^t = 10000 × (1.1)^t

Step 2: Calculate for t = 3 years

For Simple Interest:

A_SI(3) = 10000 + 1000 × 3 = 10000 + 3000 = Rs. 13,000.

For Compound Interest:

A_CI(3) = 10000 × (1.1)³ = 10000 × 1.331 = Rs. 13,310.

Step 3: Comparison

Difference = A_CI(3) - A_SI(3) = 13310 - 13000 = Rs. 310.

Conclusion: Compound Interest yields Rs. 310 more than Simple Interest after 3 years.

Question 5: Discuss the advantages and limitations of Mathematical Modelling in solving real-world problems.

Solution:

Advantages of Mathematical Modelling:

• Predictive Power: Allows us to forecast future trends (e.g., weather forecasts, population growth) without carrying out costly experiments.
• Cost & Safety: Helps test critical systems (bridges, rockets, epidemics) safely and cheaply using computer simulations.
• Clarity: Converts ambiguous real-world problems into precise mathematical statements.

Limitations of Mathematical Modelling:

• Simplifying Assumptions: Models rely on assumptions; if an assumption is incorrect, the result can be misleading.
• Complexity: Real-world situations often involve too many unpredictable variables, making exact modelling difficult.
• Context Limitations: A model valid in one situation may not work when applied to another without modification.