N2019 P1 Q11

Scientists are investigating how the temperature of water changes in various environments.

1. The scientists begin by investigating how hot water cools.
   The water is heated in a container and then placed in a room which is kept at a constant temperature of \(16\,{}^\circ \mathrm{C}\). The temperature of the water \(t\) minutes after it is placed in the room is \(\theta \,{}^\circ \mathrm{C}\mathrm{.}\) This temperature decreases at a rate proportional to the difference between the temperature of the water and the temperature of the room. The temperature of the water falls from a value of \(80\,{}^\circ \mathrm{C}\) to \(32\,{}^\circ \mathrm{C}\) in the first \(30\) minutes.
   1. Write down the differential equation for this situation. Solve this differential equation to get \(\theta \) as an exact function of \(t\).[6]
   2. Find the temperature of the water \(45\) minutes after it is placed in the room.[1]
2. The scientists then model the thickness of ice on a pond.
   In winter the surface of the water in the pond freezes. Once the thickness of the ice reaches \(3\) cm, it is safe to skate on the ice. The freezing of the water is modelled by a differential equation in which the rate of change of the thickness of ice is inversely proportional to its thickness. It is given that \(T=0\) when \(t=0.\) After \(60\) minutes, the ice is \(1\) cm thick.
   Find the time from when freezing commences until the ice is first safe to skate on.[6]