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Longhand

Before the final

Misreading Newton’s law of cooling

The habit: The law is about T minus the room temperature: the model is written for that difference, and the room is where the object ends up.

3 practice questions are built to catch it, in 1 topic.

See it fail in a figure: Decay the gap, not the temperature, in the exponential growth and decay lesson

Practise against it: 2 questions

What it looks like

A question from exponential growth and decay, with the answer this slip gives beside the right one.

Standard3 marks · 5 min

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.

With the slip
≈13.55\displaystyle \approx 13.55
Worked through
t=5log⁡7log⁡(7/4)≈17.39 minutes\displaystyle t = \frac{5\log 7}{\log(7/4)} \approx 17.39 \text{ minutes}

90ekt heads for 0 °C. The model is for the gap above the room, T − 20, which falls from 70 to 40 in 5 minutes and has to fall to 10.

See the worked solution

Where it comes up

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

  1. 01
    An object’s temperature is T(t)=24+60e−0.1tT(t) = 24 + 60e^{-0.1t} degrees Celsius after tt minutes. Which differential equation does TT satisfy, and what is the room temperature?

    Exponential growth and decayStandard

  2. 02
    A kettle of water at 100 °C is left to cool in a room of unknown constant temperature. After 10 minutes the water is at 80 °C, and after 20 minutes it is at 65 °C. Assuming Newton’s law of cooling, find the room temperature.

    Exponential growth and decayChallenging

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