Welcome to Models in Context!

Ever wondered how scientists predict the spread of a virus, or how engineers calculate the path of a rocket? They use mathematical models. In this chapter, you’ll learn how to take a real-life situation and turn it into math using the functions you’ve already studied (like linear, quadratic, and exponential functions).

Don't worry if the idea of "modelling" sounds a bit abstract at first. Essentially, we are just using math to tell a simplified story about the world around us. Let's dive in!

1. What is a Mathematical Model?

A model is a simplified representation of a real-world situation. Think of it like a map. A map isn't the actual city—it doesn't show every single blade of grass or every pebble—but it gives you enough information to get from point A to point B. Mathematical models do the same for data.

In this section of the syllabus, we focus on 1.02z: Using functions in modelling. This means taking a set of conditions and picking the right type of function to describe it.

Common Functions Used in Models

  • Linear Functions: \( y = mx + c \). Used when something changes at a constant rate (e.g., a taxi that charges a fixed fee plus a set amount per mile).
  • Quadratic Functions: \( y = ax^2 + bx + c \). Often used for projectiles (things thrown in the air) or calculating maximum profit.
  • Exponential Functions: \( y = Ab^x \) or \( y = Ae^{kx} \). Used for rapid growth (bacteria) or decay (radioactive material).
  • Trigonometric Functions: \( y = a \sin(bx + c) \). Used for periodic behavior like tides, seasons, or sound waves.
Quick Review:

If the change is a constant amount, go Linear.
If the change is a constant percentage or multiplier, go Exponential.
If it goes up and then down in a curve, try Quadratic.

2. The Modelling Process

When you are faced with a modelling question, follow these steps:

  1. Define your variables: Usually, time \( t \) is the independent variable (on the x-axis).
  2. Identify the function type: Look for keywords like "constant rate" or "proportional to."
  3. Find the constants: Use the information given (initial values) to find things like the gradient \( m \) or the starting value \( A \).
  4. Solve and Interpret: Use your equation to predict a value, then explain what that value means in the "real world."
Example: A water tank starts with 500 liters and leaks 5 liters every hour.

1. Variables: \( V \) = Volume, \( t \) = time in hours.
2. Function: Constant rate of decrease means Linear.
3. Equation: \( V = -5t + 500 \).
4. Interpretation: When \( t = 100 \), \( V = 0 \). The tank is empty after 100 hours.

3. Assumptions and Limitations

This is a favorite topic for exam questions! No model is perfect because the real world is messy. To make the math work, we have to make assumptions.

Modelling Assumptions

An assumption is something we "pretend" is true to simplify the calculation. Common ones include:

  • Ignoring air resistance when throwing a ball.
  • Assuming constant growth (e.g., a population grows by exactly 2% every year without fail).
  • Treating an object as a particle (ignoring its size and shape).

Modelling Limitations

A limitation is a reason why the model might fail or stop being accurate. Every model has a "breaking point."

Did you know? An exponential model for population growth might predict that in 500 years, there will be more humans than atoms in the universe! This is a clear limitation—the model doesn't account for limited food or space.

Common Exam Question: "State one limitation of this model."

Tip: Look at very large values of time. Does the model say the temperature of a coffee cup will eventually reach \(-1000\) degrees? That’s a limitation because it should stop at room temperature!

4. Refining and Comparing Models

Sometimes one model isn't good enough, and we need to refine it or compare it to another.

Refining a Model

To refine a model means to make it more realistic by removing an assumption. Example: If your model for a falling object is \( s = 4.9t^2 \), you might refine it by adding a term for air resistance to make it more accurate at high speeds.

Comparing Models

If you have two different models for the same data, you compare them by looking at:

  • Accuracy: Which model's predicted values are closer to the actual observed data?
  • Range of Validity: Does one model work for a long time, while the other only works for the first few minutes?
  • Simplicity: Sometimes a slightly less accurate linear model is better than a very complex cubic model because it is easier to use.
Key Takeaway:

A "good" model strikes a balance between being simple enough to calculate and accurate enough to be useful.

5. Common Mistakes to Avoid

  • Ignoring Units: If time is in minutes but the rate is "per hour," your model will be wrong. Always check your units!
  • Senseless Predictions: If your model predicts a negative height or a population of 2.5 people, recognize that the model has reached its limit.
  • Confusing "Assumptions" and "Limitations": An assumption is what you do before the math (to simplify it). A limitation is the result of those simplifications (why it's not perfectly accurate).

Summary Checklist

1. Identify the function: Is it linear, quadratic, or exponential?
2. Solve for constants: Use "Initial conditions" (usually when \( t = 0 \)).
3. Evaluate assumptions: What did you ignore (e.g., friction, varying rates)?
4. Check for limitations: Does the model produce impossible results for very large or very small values?
5. Consider refinements: How could you make the model more realistic?

Keep practicing! Modelling can feel "vague" compared to pure algebra, but once you start seeing the functions as real-world stories, it becomes one of the most rewarding parts of Mathematics A.