7. A tank initially contains 200 litres of liquid A. Liquid B is pumped into the tank at the rate of 10 litres per minute. The mixture is stirred continuously and the tank is kept full at 200 litres at all times. The contents of the tank is pumped out at the same rate of 10 litres per minute. Let X (t) denote the amount (in litres) of liquid B in the tank after t minutes. (a) Write down the differential equation for X (t), and its initial value. (b) Solve the initial value problem in (a) to find the solution X (t) for all t. (c) How long does it take until the tank contains the same amount of both liquid A and liquid B? [15] The population of a town was 100 000 in year 2000 and 60 000 in year 2002. The population is assumed to obey the Malthusian model. (a) Find the growth constant k. (b) When will the population be 10 000? (c) According to the model, what will the population be in year 2010? (d) Write down an expression for P (t), the size of the population after t years, if t = 0 in year 2010. [15]