2015 SRJC P2 Q2

A tank contains water which is heated by an electric water heater working under the action of a thermostat. When the water heater is first switched on, the temperature of the water is \(35\mathrm{ }{}^\circ \mathrm{C}\). The heater causes the temperature to increase at a rate \(r\mathrm{ }{}^\circ \mathrm{C}\) per minute, where \(r\) is a constant, until the water temperature hits \(75\mathrm{ }{}^\circ \mathrm{C}\). The heater then switches off.

1. Write down, in terms of \(r\), the time taken for the temperature to increase from \(\mathrm{35 }{}^\circ \mathrm{C}\) to \(75\mathrm{ }{}^\circ \mathrm{C}\).[1]

The temperature of the water then immediately starts to decrease. The temperature of the water at time \(t\) minutes after the heater is switched off is \(\theta \mathrm{ }{}^\circ \mathrm{C}\). It is known that the temperature of the water decreases at a variable rate \(k\left( \theta -25 \right)\mathrm{ }{}^\circ \mathrm{C}\) per minute, where \(k\) is a positive constant, until \(\theta =35\).

2. Write down a differential equation involving \(\theta \) and \(t\), to represent the situation as the temperature is decreasing.[1]
3. Given that when \(\theta =55\), the temperature is decreasing at a rate of \(5\mathrm{ }{}^\circ \mathrm{C}\) per minute, find the total length of time for the temperature to increase from \(35\ {}^\circ \mathrm{C}\) to \(75\ {}^\circ \mathrm{C}\) and then decrease to \(35\ {}^\circ \mathrm{C}\), leaving your answer in exact form, in terms of \(r\).[7]