Extrapolation means using a model to predict outside the range of the data it was built from. It can fail because the model only describes the pattern you observed, and it says nothing about what happens once something in the situation changes. A good answer gives the prediction, then states whether it is reasonable and why.
This is the last lesson in International Mathematics modelling. It builds on comparing predictions with observations.
How do you decide whether a prediction is reliable?
- Find the data range and check whether the input is inside it (interpolation) or outside it (extrapolation).
- Check the output for sense. Negative heights, volumes above the capacity and probabilities above 1 are signs of failure.
- Ask what would have to stay true for the pattern to continue, such as a constant rate.
- Write one reason that applies to the situation, and say how far from the data the prediction is.
Worked example
A candle is 20 cm tall when lit. Readings over the first 6 hours fit the model h = 20 − 1.5t, where h is the height in cm and t is the time in hours. Use the model for t = 10 and t = 20, and comment.
Step 1, data range: the readings cover 0 ≤ t ≤ 6. Both requested times are outside, so both are extrapolations.
Step 2, t = 10: h = 20 − 1.5 × 10 = 20 − 15 = 5 cm. This is a sensible height, but it assumes the candle keeps burning at the same rate.
Step 3, t = 20: h = 20 − 1.5 × 20 = 20 − 30 = −10 cm. A height cannot be negative, so this value is impossible.
Step 4, find the limit: h = 0 when 1.5t = 20, so t = 20 ÷ 1.5 = 13.33…, about 13.3 hours. The model can only apply for 0 ≤ t ≤ 13.3. After that the candle has burned out and the height is 0.
The honest conclusion is that the model gives a believable value at 10 hours, and a wrong one at 20 hours because it ignores the physical limit.
The mistake to watch for
A common slip is to report the model’s output without checking it.
Mistaken answer: “After 20 hours the candle will be −10 cm tall.”
A negative height has no physical meaning.
The correction is to state that the model stops applying when h reaches 0 at about 13.3 hours, and that after this time the candle height is 0. The equation is still the same, but the real candle no longer follows it.
Check yourself
Try these on paper, then open each answer.
1. A plant model is h = 12 + 2.5w, built from readings for w = 1 to 8. Predict the height at w = 60 and comment.
Show answer
h = 12 + 2.5 × 60 = 12 + 150 = 162 cm. This is an extrapolation far beyond 8 weeks. Plants stop growing and their growth rate changes, so the assumption of constant growth is unlikely to hold and the prediction is doubtful.
2. A prepaid balance is modelled by B = 40 − 2.5d. Use it at d = 20 and state the range of days for which it applies.
Show answer
B = 40 − 2.5 × 20 = 40 − 50 = −10. A balance of −RM10 is not possible with prepaid credit. The balance reaches 0 at d = 40 ÷ 2.5 = 16, so the model applies for 0 ≤ d ≤ 16.
3. A taxi model was built from trips of 2 km to 10 km. Which of these predictions are interpolation: 6 km, 15 km, 1 km?
Show answer
Only 6 km is inside the range 2 to 10, so it is interpolation. 15 km and 1 km are outside the range, so they are extrapolation.
Where this leads next
You now have the full set of modelling steps, from defining variables and assumptions to judging predictions. Test them all with the modelling practice set. The non-calculator working trainer can check the arithmetic in your limits.
Some students calculate the extrapolated value correctly and stop there. Our teachers coach you to add the one sentence of judgement in online one-to-one Mathematics tuition.