- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
Exponential growth and decay practice question 6
A mug of tea at 90 °C is left in a room held at 20 °C. After 5 minutes it has cooled to 60 °C. Assuming Newton’s law of cooling, when will the tea reach 30 °C?
Give the time, in minutes after the tea was poured.
Under Newton’s law of cooling, which quantity decays exponentially: the temperature itself, or something else? What is that quantity at the start?
Newton’s law says the temperature difference decays exponentially: .
Use to find , then solve .
An exact expression, or a decimal to two decimal places.
Theory UBC Math 100 §11.4CLP-1 §3.3Key results
Worked solution
Strategy
The quantity that decays exponentially is the difference between the tea and the room, not the temperature itself. Fit the constant from the 5-minute reading, then solve for the target difference.
Working
Initial difference 70 °C.
Since 4/7 < 1, k is negative: the gap to room temperature shrinks.
log(1/7) = −log 7 and log(4/7) = −log(7/4), so the signs cancel. The exact form is the answer.
Answer
The answer is 5 log 7 over log of 7/4 minutes, about 17.39 minutes.
Check
Every 5 minutes the gap to the room shrinks by the factor 4/7: 70, 40, about 22.9, 13.1, 7.5, so it passes 10 between 15 and 20 minutes.
Where this goes wrong
Modelling the temperature itself as 90ekt, which decays towards 0 °C rather than towards room temperature. The model must be written for T − 20.
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