Question $1$: $16$ Marks A metal cube is dropped into a tank of water, the temperature of which is held at $\backslash (8^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The initial temperature of the cube is $\backslash (130^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ and after $30$ minutes the temperature of the cube has dropped to $\backslash (65^\{\backslash$circ$\}\backslash ).$ Assume that Newton's law of cooling applies. $(1\mathrm{.}1)$ Write down the differential equation for the temperature of the cube, $\backslash (\backslash$mathrm$\{$T$\}(\backslash$mathrm$\{$t$\})\backslash ).(1\mathrm{.}2)$ What will the temperature of the cube be after $1$ hour? $(1\mathrm{.}3)$ Will the temperature of the cube ever reach $\backslash (1^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )?$ Justify your answer! $(1\mathrm{.}4)$ What is the initial rate of change of the temperature of the cube? Question $2$: $28$ Marks A tank with volume W litres is used to dissolve a chemical in water. At time $\backslash ($ t$=0\backslash )$ the tank contains $\backslash ($ M$\_\{0\}\backslash$mathrm$\{$~kg$\}\backslash )$ of the chemical. Water containing $\backslash ($ D $\backslash$mathrm$\{$~kg$\}\backslash )$ of the chemical per litre flows into the tank at a rate of B litres per minute. The mixture in the tank is stirred thoroughly and the tank is kept full at all times. The mixture is pumped out at a rate of $\backslash ($ B $\backslash )$ litres per minute. Let $\backslash ($ Y$($t$)\backslash )$ denote the concentration of the chemical $($in $\backslash (\backslash$mathrm$\{$kg$\}/\backslash )$ litre$)$ in the mixture leaving the tank at time $\backslash ($ t $\backslash ).(2\mathrm{.}1)$ Derive a differential equation for $\backslash ($ Y$($t$)\backslash ).(2\mathrm{.}2)$ Find the equilibrium points and draw the phase line of the model. State whether the equilibrium points are stable or not. $(2\mathrm{.}3)$ Make a rough sketch of $\backslash ($ Y$($t$)\backslash )$ as a function of $\backslash ($ t $\backslash ).$ Give examples of solution curves for all the possible choices of $\backslash ($ M$\_\{0\}\backslash ).(2\mathrm{.}4)$ Explain what will happen to $\backslash ($ Y$($t$)\backslash )$ when $\backslash ($ t $\backslash$rightarrow $\backslash$infty $\backslash ).$