1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The temperature, \(\theta ^ { \circ } \mathrm { C }\), of a car engine, \(t\) minutes after the engine is turned off, is modelled by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 15 ) ^ { 2 }$$ where \(k\) is a constant.
Given that the temperature of the car engine

• is \(85 ^ { \circ } \mathrm { C }\) at the instant the engine is turned off
• is \(40 ^ { \circ } \mathrm { C }\) exactly 10 minutes after the engine is turned off
  1. solve the differential equation to show that, according to the model

$$\theta = \frac { a t + b } { c t + d }$$ where \(a , b , c\) and \(d\) are integers to be found.

Hence find, according to the model, the time taken for the temperature of the car engine to reach \(20 ^ { \circ } \mathrm { C }\). Give your answer to the nearest minute.