Complete Solutions and Summary of Introduction to Mathematical Modelling – NCERT Class 9, Mathematics, Appendix 2 – Summary, Questions, Answers, Extra Questions
Detailed summary and explanation of Appendix 2 ‘Introduction to Mathematical Modelling’ including word problems, formulation, solution, interpretation, validation, advantages, and limitations of modelling with question answers and extra exercises from NCERT Class IX, Mathematics.
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Introduction to Mathematical Modelling
Appendix 2: Mathematics - Complete Study Guide
Chapter Overview
Formulation
Problem Setup
Solution
Math Solving
Interpretation
Real Meaning
Validation
Reality Check
What You'll Learn
Mathematical Model
Relation describing real-life situations.
Word Problems Review
Steps in solving real-world problems mathematically.
Modelling Process
Formulation, solution, interpretation, validation.
Advantages & Limits
Benefits and constraints of models.
Key Highlights
Mathematical modelling translates real-world problems into math for solutions. Used in satellite launches, monsoon predictions, pollution control, traffic management. Involves formulation (relevant factors, equations), solving, interpreting, validating. Advantages: Cheap, safe experiments. Limitations: Approximations, not exact reality.
Comprehensive Chapter Summary
1. Introduction to Mathematical Modelling
- From earlier classes, solving real-world problems like simple interest using formulas.
- Formula is a mathematical model describing relations between quantities.
- Models used for satellite launches, monsoon prediction, pollution control, traffic jams.
- Process: Take real problem, formulate mathematically, solve, interpret, validate.
- Expanded: Models simplify complex realities for analysis and prediction. Essential in science, engineering, economics.
- Historical context: Evolved from basic equations to complex simulations.
- Importance: Allows testing scenarios without real-world risks or costs.
- Examples illustrate transition from word problems to full modelling.
Real-Life Applications
Launching satellites, predicting monsoons, controlling vehicle pollution, reducing traffic jams in cities.
2. Review of Word Problems
- Steps: Formulation (identify relation, equation), solution, interpretation.
- Example 1: Petrol for distance, direct variation \( y = kx \), assumptions constant conditions.
- Example 2: Simple interest \( I = \frac{Pnr}{100} \), find time, assume constant rate/price.
- Example 3: Boat speeds upstream/downstream, equations for distance, assume constant speeds/friction negligible.
- Expanded: Word problems build foundation for modelling. Identify relevant/irrelevant factors.
- Assumptions crucial: Ignore minor effects for simplicity.
- Mathematical description key: Translate words to equations.
- Interpretation links math back to reality.
Word Problem Steps
- Formulation: Relevant factors, equation.
- Solution: Solve math problem.
- Interpretation: Apply to real context.
Assumptions in Problems
- Constant rates/speeds.
- Neglect friction/variations.
Exercise A2.1 Problems
- Computer hire vs buy.
- Cars meeting time.
- Moon orbital speed.
- Water heater hours.
3. Some Mathematical Models
- Add validation step: Check solution against reality, modify if needed.
- Example 4: Room tiling, calculate tiles, validate practical (extra for damage).
- Example 5: Girls enrolment, linear model, validate errors, revise with correction.
- Population growth: Exponential model \( P = P_0 e^{rt} \).
- Radioactive decay: \( A = A_0 e^{-kt} \).
- Newton's cooling: \( T - T_s = (T_0 - T_s) e^{-kt} \).
- Expanded: Validation ensures model accuracy. Iterative process improves fit.
- Models approximate; balance simplicity and precision.
- Applications: Social issues like gender equality, scientific phenomena.
- Exercise A2.2: 400m race timings model.
Girls Enrolment Model
Data from 1991-2002, linear increase 0.22%/year, predict 50% in 2027 after revision.
Population Growth
India 2001-2011 data, model \( P = 1027 e^{0.014t} \), predict 2026 population.
4. The Process of Modelling, Advantages and Limitations
- Formulation: State problem, relevant factors, math description.
- Solution: Solve equations.
- Interpretation: Relate to real-world.
- Validation: Check/match reality, revise.
- Advantages: Cheap, safe, quick experiments; predict/explain.
- Limitations: Approximations, not exact; apply within limits.
- Expanded: Modelling iterative, creative. Balances accuracy/ease.
- Examples: Taj corrosion, school needs, motion laws.
- Exercise A2.3: Traffic waiting, factors.
- Summary points reinforce key ideas.
Advantages
Model Taj corrosion without risk, predict schools mathematically.
Key Concepts and Definitions
Mathematical Model
Math relation for real situations.
Formulation
Problem to math equations.
Solution
Solve math problem.
Interpretation
Apply solution to reality.
Validation
Check against real data.
Relevant Factors
Important variables.
Assumptions
Hypotheses for simplicity.
Important Facts
Linear
Model
Exponential
Growth
Decay
Radioactive
Cooling
Newton
Validation
Key
Questions and Answers from Chapter
Short Questions (1 Mark)
Q1. What is a mathematical model?
Answer: Math relation for real situations.
Q2. Name one use of models.
Answer: Predicting monsoons.
Q3. What is formulation?
Answer: Problem to math.
Q4. What is validation?
Answer: Check reality.
Q5. In petrol example, what varies directly?
Answer: Petrol and distance.
Q6. Simple interest formula?
Answer: \( I = \frac{Pnr}{100} \).
Q7. Boat upstream speed?
Answer: x - 2.
Q8. Assumption in boat problem?
Answer: Constant speeds.
Q9. Tiles example, number along length?
Answer: 20.
Q10. Enrolment initial year?
Answer: 41.9%.
Q11. Population growth formula?
Answer: \( P = P_0 e^{rt} \).
Q12. Decay constant k?
Answer: Positive.
Q13. Cooling law?
Answer: Exponential.
Q14. Advantage of modelling?
Answer: Cheap experiments.
Q15. Limitation?
Answer: Approximations.
Q16. Relevant factors in formulation?
Answer: Important ones.
Q17. In computer hire, what to find?
Answer: Months.
Q18. Cars speed sum?
Answer: 70 km/h.
Q19. Moon distance?
Answer: 384000 km.
Q20. Heater cost per hour?
Answer: Rs 8.
Medium Questions (3 Marks)
Q1. In petrol example, find k.
Answer: \( k = \frac{48}{432} = \frac{1}{9} \). Use for new distance.
Q2. Interest example, equation for n.
Answer: 4000 = 1200n, n=10/3 years.
Q3. Boat equation.
Answer: 5(x+2)=6(x-2), x=22 km/h.
Q4. Tiles formula.
Answer: (20*16)+20=340 tiles.
Q5. Enrolment model (2).
Answer: 50=41.9+0.22n, n≈37.
Q6. Revised enrolment.
Answer: 50=42.08+0.22n, n=36.
Q7. Population 2026.
Answer: Use \( P=1027 e^{0.014*25} \approx 1457 \).
Q8. Decay half-life.
Answer: Time for A=A0/2.
Q9. Cooling solution.
Answer: T=Ts + (T0-Ts)e^{-kt}.
Q10. Computer months.
Answer: 25000=2000n, n=12.5 months.
Q11. Cars meeting time.
Answer: 100/(40+30)≈1.43 hours.
Q12. Moon speed.
Answer: Circumference/24 hours.
Q13. Heater hours.
Answer: (1240-1000)/8=30 hours/day? Wait, per month.
Q14. Race model.
Answer: Linear decrease in timings.
Q15. Traffic factors.
Answer: Arrival rates, not price.
Q16. Relevant in petrol.
Answer: Distance, petrol; ignore route.
Q17. Assumptions in interest.
Answer: Constant rate, price.
Q18. Boat assumptions.
Answer: Constant speeds, negligible friction.
Q19. Tiles validation.
Answer: Add extra for damage.
Q20. Enrolment validation.
Answer: Check errors, revise.
Long Questions (6 Marks)
Q1. Suppose a company needs a computer for some period of time. The company can either hire a computer for Rs 2,000 per month or buy one for Rs 25,000. Find the number of months beyond which it will be cheaper to buy a computer.
Answer: Formulation: Relevant - hire cost, buy cost; irrelevant - others. Equation: 2000n > 25000. Solution: n > 12.5. Interpretation: After 13 months cheaper to buy. Validation: Assumes constant use, no maintenance.
Q2. A car starts from a place A and travels at a speed of 40 km/h towards another place B. At the same instance, another car starts from B and travels towards A at a speed of 30 km/h. If the distance between A and B is 100 km, after how much time will the cars meet?
Answer: Formulation: Relative speed 70 km/h. Equation: t=100/70. Solution: t≈1.43 hours. Interpretation: Meet after 1 hour 26 min. Validation: Constant speeds, straight path.
Q3. The moon is about 3,84,000 km from the earth, and its path around the earth is nearly circular. Find the speed at which it orbits the earth, assuming that it orbits the earth in 24 hours. (Use π = 3.14)
Answer: Formulation: Circumference 2πr. Equation: Speed = 2*3.14*384000 / 24. Solution: ≈1005 km/h. Interpretation: Orbital speed. Validation: Approximate circular, sidereal day actually 27.3 days.
Q4. A family pays Rs 1000 for electricity on an average in those months in which it does not use a water heater. In the months in which it uses a water heater, the average electricity bill is Rs 1240. The cost of using the water heater is Rs 8.00 per hour. Find the average number of hours the water heater is used in a day.
Answer: Formulation: Difference 240 = 8*h*30. Equation: h=240/(8*30)=1. Solution: 1 hour/day. Interpretation: Average use. Validation: Assumes 30 days, constant cost.
Q5. Using 400m race data, construct model, estimate next Olympics timing.
Answer: Formulation: Linear decrease. Find rate from data. Solution: Average decrease ~0.3s/year. From 2004 49.41, 2028 ~47s. Interpretation: Predicted time. Validation: Check past, revise if nonlinear.
Q6. Suppose you want to minimise the waiting time of vehicles at a traffic junction of four roads. Which factors are important?
Answer: Formulation: Arrival rates, signal timings important; price irrelevant. Model queue theory. Solution: Optimize cycles. Interpretation: Reduce waits. Validation: Simulate traffic.
Q7. I travelled 432 km on 48 litres petrol. How much for 180 km?
Answer: Formulation: Direct variation y= (48/432)x. Solution: 20 litres. Interpretation: Needed petrol. Validation: Same conditions.
Q8. Invest Rs 15000 at 8%, buy Rs 19000 machine. Find period.
Answer: Formulation: I=4000= (15000*8*n)/100. Solution: n=10/3 years. Interpretation: 3 years 4 months. Validation: Constant rate.
Q9. Boat upstream 6h, downstream 5h, stream 2 km/h. Find still speed.
Answer: Formulation: d=6(x-2)=5(x+2). Solution: x=22 km/h. Interpretation: Still water speed. Validation: Constant flow.
Q10. Room 6m x 5m, tiles 30cm. How many tiles?
Answer: Formulation: 20 along length, 16 breadth +20. Solution: 340. Interpretation: Total tiles. Validation: Extra for cuts.
Q11. Girls enrolment data, model to 50%.
Answer: Formulation: Linear 41.9+0.22n. Solution: n=37. Interpretation: 2028. Validation: Revise to 42.08+0.22n, n=36, 2027.
Q12. Population 1027 million 2001, rate 1.4%. 2026 population.
Answer: Formulation: P=1027 e^{0.014*25}. Solution: ≈1457 million. Interpretation: Projected. Validation: Check trends.
Q13. Radioactive decay model.
Answer: Formulation: A=A0 e^{-kt}. Solution: For half-life, t=(ln2)/k. Interpretation: Amount left. Validation: Lab data.
Q14. Newton's cooling for coffee.
Answer: Formulation: T=Ts+(T0-Ts)e^{-kt}. Solution: Solve for t. Interpretation: Time to cool. Validation: Measure.
Q15. Discuss modelling process steps.
Answer: Formulation: State, factors, description. Solution: Solve. Interpretation: Real meaning. Validation: Check, revise.
Q16. Advantages of modelling.
Answer: Cheap, safe, quick; predict without experiments. Examples: Taj corrosion, schools need.
Q17. Limitations of modelling.
Answer: Approximations, not exact; apply within limits. Like map vs country.
Q18. How word problems differ from modelling?
Answer: Modelling adds validation, iteration; real-life complex. Word problems simplified.
Q19. In traffic model, important factors?
Answer: Arrival rates, signal times; irrelevant price.
Q20. Balance in good model?
Answer: Accuracy and ease of use. Simple yet effective like Newton's laws.
Interactive Knowledge Quiz
Test your understanding of Mathematical Modelling
Quick Revision Notes
Modelling Steps
- Formulation
- Solution
- Interpretation
- Validation
Models Types
- Linear
- Exponential
- Decay
Advantages
- Cheap
- Safe
Exam Strategy Tips
- Identify factors
- Form equations
- Solve interpret
- Validate models
- List assumptions
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