- A cup of hot tea was placed on a table. At time \(t\) minutes after the cup was placed on the table, the temperature of the tea in the cup, \(\theta ^ { \circ } \mathrm { C }\), is modelled by the equation
$$\theta = 25 + A \mathrm { e } ^ { - 0.03 t }$$
where \(A\) is a constant.
The temperature of the tea was \(75 ^ { \circ } \mathrm { C }\) when the cup was placed on the table.
- Find a complete equation for the model.
- Use the model to find the time taken for the tea to cool from \(75 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\), giving your answer in minutes to one decimal place.
Two hours after the cup was placed on the table, the temperature of the tea was measured as \(20.3 ^ { \circ } \mathrm { C }\).
Using this information,
- evaluate the model, explaining your reasoning.