Exercises 3.2findx(t). We can determine the concentration of salt in the first tank by dividingx(t) by theits volume, i.e.,x(t)/60 kg/gal. Note that the volume of brine in this tank remains constantbecause the ﬂow rate in is the same as the ﬂow rate out. Thenoutput rate1= (3 gal/min)·x(t)60kg/gal=x(t)20kg/min.Since the incoming liquid is pure water, we conclude thatinput rate1= 0.Therefore,x(t) satisfies the initial value problemdxdt= input rate1−output rate1=−x20,x(0) =x0.This equation is linear and separable. Solving and using the initial condition to evaluate thearbitrary constant, we findx(t) =x0e−t/20.Now, lety(t) denote the mass of salt in the second tank at time
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