1. Narrow the question
Instead of “the mathematics of coffee”, investigate: How well does Newton’s law of cooling model the temperature of a hot drink over its first 20 minutes in a room at 22°C?
This question defines a measurable outcome, a model and an observation period. In a real investigation, record the ambient temperature and explain how the thermometer, sampling interval and container affect the measurements.
2. Define and use the model
If the room temperature remains constant and the cooling rate is proportional to the temperature difference, a candidate model is:
Here, T is temperature in °C, t is time in minutes, and k is a positive constant in min⁻¹. The model starts at 82°C. Using an illustrative measurement of 62°C after 5 minutes:
The resulting prediction at 10 minutes is approximately 48.7°C. If an illustrative observation is 50.0°C, the residual (observed minus predicted) is about +1.3°C.
3. Evaluate the model
A plot of residuals against time can reveal a systematic mismatch. Mostly positive residuals late in the experiment, for example, would suggest the model predicts temperatures that are too low in that period. Random-looking residuals alone do not prove the assumptions are correct.
Compare the size of the errors with measurement uncertainty, and discuss possible changes in ambient temperature, heat loss through the container and evaporation. Do not assume a model fitted over 20 minutes remains equally useful outside that interval.
4. Before applying this to your IA
- Choose a question you can investigate with your own evidence.
- Explain each variable, assumption and fitting choice.
- Include graphs and enough observations to evaluate the model.
- Connect the mathematics and depth of analysis to your course and level.
This short excerpt illustrates reasoning and revision. It is not a complete IA and does not demonstrate every assessment criterion.
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