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Modelling on Differential..
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A tank contains 100 litres of salt solution, in which 20kg of salt are dissolved. Beginning at time t=0, pure water runs into this tank at a rate of 5 litres/minute. The mixture flows out into a second large tank at the same rate: the second tank initially contains 100 litres of pure water. Let x_i(t) represent the... 顯示更多 A tank contains 100 litres of salt solution, in which 20kg of salt are dissolved. Beginning at time t=0, pure water runs into this tank at a rate of 5 litres/minute. The mixture flows out into a second large tank at the same rate: the second tank initially contains 100 litres of pure water. Let x_i(t) represent the amount of salt(in kg) in tank i at time t; let c_i(t) be the concentration of the salt solution(in kg/litre) in tank i at time t and let V_i(t) be the volume (in litres) of solution in tank i at time t. Here, i = 1 for the first tank and i = 2 for the second tank. (i) Write down an equation for the volume V_i(t) of solution in each tank at time t. (ii) Write down a first initial value problem for x_1(t), and hence determine c_1(t) for t > 0. (iii) Write down a first order initial value problem for x_2(t), and hence determine c_2(t) for t >0. (iv) Let t = T be the time at which c_2(t) has its largest value. Show that 2 + s = e^s, where s =T/20. Hence find an approximate value for T, correct to the nearest 2 minutes.
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