- The temperature, \(\theta ^ { \circ } \mathrm { C }\), of coffee in a cup, \(t\) minutes after the cup of coffee is put in a room, is modelled by the differential equation
$$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 20 )$$
where \(k\) is a constant.
The coffee has an initial temperature of \(80 ^ { \circ } \mathrm { C }\)
Using \(k = 0.1\)
- use two iterations of the approximation formula \(\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { 0 } = \frac { y _ { 1 } - y _ { 0 } } { h }\) to estimate the temperature of the coffee 3 minutes after it was put in the room.
The coffee in a different cup, which also had an initial temperature of \(80 ^ { \circ } \mathrm { C }\) when it was put in the room, cools more slowly.
- Use this information to suggest how the value of \(k\) would need to be changed in the model.
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