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

(i) 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 $\theta^{\circ} \mathrm{C}$. 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.

(a) Write down a differential equation for this situation. Solve this differential equation to get $\theta$ as an exact function of $t$.

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(b) Find the temperature of the water 45 minutes after it is placed in the room.

(ii) The scientists then model the thickness of ice on a pond.

[1]

In winter the surface of the water in the pond freezes. Once the thickness of the ice reaches $3 \mathrm{~cm}$, it is safe to skate on the ice. The thickness of the ice is $T \mathrm{~cm}, t$ minutes after the water starts to freeze. The freezing of the water is modelled by a differential equation in which the rate of change of the thickness of the ice is inversely proportional to its thickness. It is given that $T=0$ when $t=0$. After 60 minutes, the ice is $1 \mathrm{~cm}$ thick.

Find the time from when freezing commences until the ice is first safe to skate on.

[6]