2. You make a cup of Yorkshire tea at 100C and put it in a room at 20C. You plan to add 20% milk (regular or soya), at 5C when added. You want to know if the finished drink will be cooler if you either: i. add the milk immediately after boiling the water, then wait 5 minutes (300s); or ii. wait 5 minutes (300s) for the hot tea to cool and then add the milk. Model the situation above to compare both cases and calculate the answer. For simplicity assume that the total rate of temperature decay is governed by the coefficient k = UA = 0.001/s. (10 marks] (a) Write a mathematical model in terms of an ordinary differential equation and its solution (both general and particular for the two cases), stating what each component means. Say which case results in the cooler drink. (b) The hot drink is a simple model of a building losing heat to its surroundings, but the reality might be more complicated, such as in the schematic below: T ui U2 Outside U3 Assume that heat can flow equally well across any of the partitions and walls, and that conductivities and thermal masses are the same for each surface and room. Hence write a system of three differential equations for the rates of change of the temperatures ui, uz, uz, then rearrange into vector / matrix form. [10 marks] [5 marks] (c) Find the General Solution to the heat flow equation with sinusoidally varying external temperature: du + ku = A cos(wt) dt Use the expression cos(x) = Re(ext) to simplify the maths