8.1 Mathematical Modeling
What is Mathematical Modeling?
Mathematical modeling is the process of using mathematical concepts, structures, and language to represent real-world situations, analyze them, and make predictions or informed decisions.
The 6 Steps of the Mathematical Modeling Process
Mathematical modeling is an iterative cyclical process consisting of six fundamental steps:
- Identifying and Defining the Problem: Clearly state the real-world problem, identify key variables, and determine the objectives of the model.
- Making Assumptions and Identifying Variables: Simplify the real-world complexity by establishing reasonable assumptions and classifying variables into independent and dependent variables.
- Applying Mathematics to Solve Problems: Translate the real-world situation into mathematical equations, functions, or geometric models, and use algebraic or computational tools to find mathematical solutions.
- Interpreting Results: Convert mathematical solutions back into real-world context and evaluate what the solutions mean practically.
- Validating the Model: Compare predictions or results generated by the model against real-world data or physical constraints. If results are inaccurate or unrealistic, revisit assumptions and refine the model.
- Reporting the Findings: Document the model, methodology, assumptions, limitations, and conclusions clearly for decision-making.
8.2 Types of Mathematical Models
1. Linear Models
Used when the rate of change between two variables is constant:
$$y = mx + c$$
- Example: Calculating total taxi fare based on a fixed baseline charge plus a constant charge per kilometer.
2. Quadratic Models
Used when relationships involve acceleration, parabolic trajectories, or maximum/minimum optimization problems:
$$y = ax^2 + bx + c \quad (a \neq 0)$$
- Example: Modeling the motion of a projectile (e.g., throwing a ball or launching a rocket) over time $t$.
3. Exponential Models
Used for situations involving rapid growth or decay proportional to the current amount:
$$y = a \cdot b^x \quad \text{or} \quad y = a e^{kx}$$
- Example: Compound interest growth, population expansion, or radioactive decay over time.
Evaluating and Refinding Models
Real-world mathematical models are rarely perfect on the first attempt. Limitations occur due to simplified assumptions. If new parameters or changing conditions arise, the model must be adjusted iteratively.