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4. (a) Explain the idea behind Newton's Law of Cooling, as denoted in the following equation, which models the temperature change for a room. dT dt -= k (Te - T) Assume T is the room temperature, Te is the outside temperature, and k is a positive constant (cooling constant). Extend this equation so that it can model heat generation controlled by a thermostat. The thermostat is set with a target temperature Tg and a positive constant g is used to model the time taken for heat to be generated. [5 marks] (b) Write a MATLAB function (m-file), which is called from ode450, to implement the model from (a), with the following initial conditions and constants. Teis -10C T starts at 200 Tois 20C k=0.10,g=0.75 Time runs from 0 to 24 (hours) Furthermore, assume that the heat is switched on for 1 hour every three hours, starting from time 0. If the heater was broken, discuss how you would approximate the room temperature after 10 hours. [14 marks) (c) Consider a species of bird that can be split into three age groupings: 0-1 (year 1), 1-2 (year 2) and 2-3 year 3). The population is observed once a year. The Leslie Matrix is as follows: 0 L = 0.3 0 3 1 0 0 Xe = 0.5 0 100 200 0 Based on this: Explain the values in the Leslie matrix (L) • Show how linear algebra can be used to predict the total population after 10 years. • What are the simplifying assumptions of the Leslie model? [6 marks)