A 280-gallon tank initially contains 100 gallons of brine in which 5 pounds of s

Question

A 280-gallon tank initially contains 100 gallons of brine in which 5 pounds of salt is dissolved. Beginning at a certain moment, when t = 0, brine containing 0.25 pounds of salt per gallon is pumped into the tank at a rate of 9 gallons/minute, and the well-mixed solution is pumped out at the rate of 6 gallons/minute. Let denote the number of minutes it takes for the tank to become full. Let Q(t) denote the number of pounds of salt in the tank at any time t, where 0 le t le t1, and t denotes the number of minutes after the brine starts being pumped into the tank. Find the following: State and then solve a first order linear differential equation and initial condition whose unique solution on [0, t1] is Q(t). After how many minutes does the tank become full, i.e., t1 = ? How many pounds of salt does the tank contain at the instant when it becomes full, i.e., Q(t1) = ?

Explanation / Answer

let the amount of salt in the tank as a function of time be s(t). so rate of change of salt with time=ds/dt ds/dt=rate(in)-rate(out) =5-6s/(100+3t) this is the required differential equation, s(t)=5(100+3t)/(9) + c/(100+3t)^2 now c can be found by putting t=0 and s(0)=5,we get c=40000/9 now u will be able to get the required results

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