Lesson Notes By Weeks and Term v3 - Senior Secondary 3
Modelling
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Modelling
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Subject: Further Mathematics
Class: Senior Secondary 3
Term: 1st Term
Week: 4
Theme: Operation Research
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Explain concept and importance of modelling Distinguish between dependent and in dependent variables in modelling Give examples of models State methodology of model building Explain solution of problems in modelling Give application to physical, biological, social and behavioural services
Definition of a Model: A model is a simplified representation of a real-world system or phenomenon. It aims to capture the essential features and relationships of the system in a more manageable form. Models can be physical, graphical, or mathematical. In Further Mathematics, the focus is predominantly on mathematical models.
Definition of Modelling: Modelling is the process of translating a real-world problem or system into a mathematical representation, analyzing this representation, and then interpreting the results back in the context of the original problem. It involves abstraction, simplification, and idealization to make complex situations tractable.
Importance of Modelling: Understanding Complex Systems: Models help to break down intricate systems into manageable components, making it easier to understand their behaviour and underlying mechanisms. For instance, understanding traffic flow in a city like Lagos or the spread of an epidemic in a community.
Prediction and Forecasting: Models can be used to predict future outcomes or behaviours based on current data and assumptions. Examples include predicting economic trends, weather patterns, or crop yields in different regions of Nigeria.
Optimization and Decision-Making: By manipulating variables within a model, decision-makers can find optimal solutions (e.g., maximizing profit, minimizing cost, allocating resources efficiently). This is crucial for businesses, government agencies, and organizations aiming for efficiency.
Cost-Effectiveness and Risk Reduction: Experimenting with real-world systems can be expensive, time-consuming, or risky. Models provide a safe and cost-effective environment to test different scenarios and strategies before implementation. For example, testing the impact of a new agricultural policy without implementing it nationwide.
Communication: Models provide a structured and quantitative way to communicate insights and findings about a problem to various stakeholders. In a mathematical model, variables are quantities that can change or vary.
Independent Variable(s): These are the variables that are controlled or changed by the modeller or are naturally occurring inputs that influence the system. They are often referred to as input variables, predictor variables, or explanatory variables. In an equation, they are typically on the right-hand side.
Dependent Variable(s): These are the variables that are measured or observed and whose values are influenced or determined by the independent variables. They are often referred to as output variables, response variables, or outcome variables. In an equation, they are typically on the left-hand side.
Example 1: Crop Yield A farmer wants to understand how the amount of fertilizer applied affects maize yield.
Independent Variable: Amount of fertilizer applied (the farmer controls this).
Dependent Variable: Maize yield (the yield depends on the fertilizer amount).
Example 2: Cost of Production A small business produces "puff-puff" (a popular Nigerian snack) and wants to model its total cost. Let `C` be the Total Cost. Let `N` be the Number of puff-puff produced. Let `V` be the Variable cost per puff-puff (flour, oil, sugar). Let `F` be the Fixed cost (rent, equipment depreciation).
The model could be: `C = V * N + F` Independent Variable: `N` (Number of puff-puff produced – this changes and affects cost).
Dependent Variable: `C` (Total Cost – this depends on the number produced). The process of building and using a mathematical model typically follows these steps: Problem Definition: Clearly understand the real-world problem or system to be modelled. Identify the objectives of the modelling exercise (what questions need to be answered? what decisions need to be made?). Define the scope and boundaries of the problem.
Example: A local government wants to optimize waste collection routes to minimize fuel costs and collection time in a specific district. Data Collection and Identification of Variables: Gather all relevant data related to the problem (e.g., historical data, current statistics). Identify the key variables that influence the system's behaviour. Distinguish between independent and dependent variables.
Example: Map of the district, locations of waste bins, capacities of waste trucks, fuel consumption rates, speed limits, labour costs, current collection schedules.
Independent variables: truck routes, number of trucks.
Dependent variables: total fuel cost, total collection time. Model Formulation (Mathematical Representation): Translate the real-world relationships and constraints into mathematical equations, inequalities, or logical statements. Make necessary assumptions and simplifications to make the model tractable, while ensuring it remains a realistic representation.
Example: Develop equations to calculate distance travelled for each route segment, fuel cost per km, time taken for collection at each point, and total time for a route. Formulate constraints for truck capacity and driver working hours. The objective function would be to minimize total cost or time.
Model Solution: Apply appropriate mathematical techniques or algorithms to solve the formulated model. This might involve algebraic manipulation, calculus, linear programming, simulation, numerical methods, or computer software.
Example: Using linear programming software to find the optimal routes that minimize the objective function subject to all identified constraints.
Model Validation and Analysis: Check if the model's predictions or solutions are logical, realistic, and accurate when compared to real-world data or expert judgment. Test the model's sensitivity to changes in input parameters (sensitivity analysis). If the model is not sufficiently accurate, refine its assumptions or structure and repeat the formulation and solution steps.
Example: Compare the model-generated optimal routes and estimated costs/times with actual historical data or pilot runs. If the model suggests a route that is impractical due to road conditions, revise the model.
Implementation and Interpretation: Translate the mathematical solution back into practical, actionable recommendations for the real-world problem. Implement the solution and monitor its effectiveness.
Example: Present the optimized waste collection routes to the local government, explain the expected savings in fuel and time, and oversee the implementation of the new routes. Solving problems in modelling involves finding the values of the dependent variables (or decisions) that satisfy the model's equations/inequalities, often with the aim of optimizing an objective function (e.g., maximizing profit, minimizing cost).
Common Solution Techniques: Algebraic Methods: For simple linear equations, simultaneous equations (substitution, elimination, matrix methods).
Calculus: For optimization problems involving continuous functions (finding maxima/minima using differentiation).
Linear Programming: For optimization problems with linear objective functions and linear inequality constraints (graphical method for 2 variables, Simplex method for more). This is very common in resource allocation.
Simulation: For complex systems where analytical solutions are difficult. Involves running computer-based experiments to observe system behaviour under different conditions.
Numerical Methods: Approximations for problems that cannot be solved analytically (e.g., finding roots of complex equations).
Interpreting the Solution: The numerical solution obtained from the model must be translated back into the context of the original real-world problem. What do the numbers mean in terms of the problem? What are the practical implications of the solution? Are there any limitations or assumptions that need to be considered when implementing the solution?
Example: A farmer wants to maximize profit from planting yam and cassava on 10 hectares of land.
Model: Let `x` be hectares of yam, `y` be hectares of cassava. `x + y = 0, y >= 0` (non-negativity) Profit from yam = N500,000/hectare, cassava = N300,000/hectare.
Objective: Maximize `P = 500,000x + 300,000y` Solution (Simplified): If no other constraints, the farmer should plant all 10 hectares with yam to maximize profit. `x=10, y=0`.
Interpretation: The farmer should allocate all available 10 hectares of land to yam cultivation to achieve the maximum possible profit of N5,000,000, assuming no other limiting factors like labour or capital for yam, or market demand for yam.
Models exist in various forms: Physical Models: Scale replicas of real objects (e.g., miniature cars, architectural models of proposed buildings in Abuja, anatomical models in biology labs).
Conceptual/Graphical Models: Diagrams, flowcharts, maps, organisation charts (e.g., a map of Nigeria showing states, a flowchart of election processes).
Mathematical Models: These use mathematical language (equations, inequalities, functions) to describe the relationships between variables. This is the primary focus.
Linear Models: `Cost = Fixed Cost + (Variable Cost per Unit Number of Units)` `Profit = (Selling Price per Unit - Variable Cost per Unit) Number of Units - Fixed Cost`
Example: A taxi driver's daily earning (`E`) might be modelled as `E = R D + T`, where `R` is the rate per km, `D` is distance travelled, and `T` is tips.
Quadratic Models: Used for problems involving curves, e.g., projectile motion. `h(t) = ut - 0.5gt^2` (height `h` at time `t` for an object thrown upwards with initial velocity `u`).
Example: Modelling the trajectory of a thrown javelin during a school sports competition.
Exponential Models: Used for growth or decay phenomena. `P(t) = P_0 e^(kt)` (population growth, where `P_0` is initial population, `k` is growth rate, `t` is time).
Example: Modelling the growth of bacteria in a laboratory or the spread of a rumour in a community.
Simultaneous Equations/Inequalities: Used to model systems with multiple constraints or interacting parts.
Example: A manufacturing company producing two types of products (e.g., Gala sausage rolls and bottled water) with limited resources (ingredients, machine hours, labour). This leads to systems of linear inequalities for resource allocation (Linear Programming).
Traffic Management in Nigerian Cities: Application: Mathematical models (e.g., queueing theory, network flow models) are used to analyze traffic patterns, predict congestion points, and optimize traffic light timings in heavily congested cities like Lagos, Abuja, and Port Harcourt.
Local Context: The models can help urban planners decide where to construct new roads, flyovers, or public transport routes to alleviate traffic bottlenecks on specific major roads (e.g., Oshodi-Apapa Expressway, Airport Road Abuja). They can also model the impact of policies like odd/even number plate restrictions. Agricultural Productivity and Food Security: Application: Modelling can predict crop yields (e.g., yam, rice, cassava) based on environmental factors (rainfall patterns, soil type, temperature), farming practices (fertilizer use, irrigation), and pest incidence.
Local Context: This helps Nigerian farmers and agricultural agencies make informed decisions about planting schedules, fertilizer application, and water management, especially with changing climate patterns. It can also inform government policies on food security, storage, and distribution to prevent scarcity and manage prices.
Resource Allocation for Public Services: Application: Governments and NGOs use models to optimize the allocation of limited resources (funds, personnel, equipment) to public services like healthcare, education, and social welfare programs.
Local Context: For instance, a state government can use models to determine the optimal placement of primary healthcare centers, schools, or boreholes in rural communities to maximize access and impact given budgetary constraints. During a health crisis (e.g., COVID-19 pandemic), models helped allocate ventilators, testing kits, and vaccination points across states.