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Question : We have two interconnected tanks. Initially tank 1 contains 30 : 82836

We have two interconnected tanks. Initially tank 1 contains 30 gallons of water and 25 oz of salt, while tank 2 initially contains 20 gallons of watter and 15 oz of salt. Water containg 1 0z/gal of salt flows into tank 1 at 1.5 gal/min. The mixture flows from tank 1 to tank 2 at 3 gal/min. Water containg 3 oz/gal of salt also flows into tank 2 at 1 gal/min. The mixture drains from tank 2 at the rate of 4 gal/min, of which some flows back into tank 1 at 1.5 gal/min.

1) Let \(Q_{1}(t)\) and \(Q_{2}(t)\) be the amount of salt in oz in each tank at time t. Write the differential equations and inital conditions that model the flow process.

2) Find the values of \(Q_{1}\) and \(Q_{2}\) for which the system is in equilibruim. let these be \(Q_{1}^{e}\) and \(Q_{2}^{e}\) .

3) Let \(x_{1} = Q_{1}(t) - Q_{1}^{e}\) and \(x_{2} = Q_{2}(t) - Q_{2}^{e}\) , determine an initial value problem for \(x^{1}\) and \(x^{2}\)

4) When do tank 1 and tank 2 reach equilibrium?