Modelling with Exponentials

This lesson brings together everything from the topic to tackle modelling problems — a key focus of the Edexcel 9MA0 specification. You will be expected to set up exponential models from context, determine constants from data, use the model to make predictions, and critically evaluate its suitability.

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The Modelling Cycle

1. Identify the appropriate model from the context
2. Set up the equation with unknown constants
3. Determine the constants using given data
4. Use the model to make predictions
5. Evaluate the model's suitability and limitations

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Setting Up Models

Growth: y = Aeᵏᵗ (k > 0)

Used when the rate of increase is proportional to the current value.

Example: A colony of insects is growing. Initially there are 200 insects. After 5 days there are 350. Assuming exponential growth, find the model.

• At t = 0: 200 = Ae⁰, so A = 200
• At t = 5: 350 = 200e⁵ᵏ
• e⁵ᵏ = 350/200 = 7/4
• 5k = ln(7/4)
• k = ln(7/4)/5 ≈ 0.1120

Model: N = 200e^(0.112t)

Decay: y = Aeᵏᵗ (k < 0) or y = Ae⁻ᵏᵗ (k > 0)