In a cooling model with a constant room temperature, the gap between the object and the room decays exponentially. If that gap halves in ten minutes, it halves again in the next ten minutes under the same model.
Define the gap before solving the differential equation. When room temperature is 20°C, write the gap as T − 20. An initial temperature of 80°C then gives an initial gap of 60°C.
- Subtract room temperature to define the gap.
- Solve for the gap or find its multiplier from a measurement.
- Add room temperature back to recover the object’s temperature.
Worked example | From 80°C to 50°C in ten minutes
Suppose dT/dt = −k(T − 20) with k positive and t measured in minutes. Write G = T − 20. Then dG/dt = −kG. For a positive gap, separating and integrating gives ln G = −kt + C.
The initial gap is 60, so G = 60e−kt. After ten minutes, T = 50 and G = 30. Thus e−10k = ½ and k = (ln 2)/10 per minute.
Use the same multiplier again
At 20 minutes the gap is 60 × (½)² = 15°C. The object’s temperature is 35°C. Each ten-minute interval halves the remaining gap. The first interval lowers the temperature by 30°C and the second lowers it by 15°C.
As time increases, the exponential term tends to zero, so T approaches 20°C. If T starts at 20°C, the constant function T = 20 also solves the original equation.
Find when the temperature first reaches 27.5°C
Use the model above. How long does it take for the temperature to reach 27.5°C?
Answer. The remaining gap is 7.5°C. Since 60 × (½)³ = 7.5, three ten-minute intervals have elapsed. The answer is 30 minutes.
Frequently asked questions
Why check the constant solution separately?
Separating the equation by dividing by G assumes G is nonzero. The case G = 0 satisfies the original differential equation directly.
Does the same halving time apply if the room temperature changes?
This model assumes constant room temperature and constant k. A changing room temperature requires a different model.
Official information
SEAB, 2027 GCE A-Level H2 Mathematics syllabus (9758), Differential Equations, formulating and interpreting solutions. Syllabus checked 11 November 2026. The examples are original.