A tank initially contains 120 liters of pure water. A salt solutions with concen

Question

A tank initially contains 120 liters of pure water. A salt solutions with concentration of 4 grams/liter of salt enters the tank at a rate of 2 liters/min and the well-stirred mixture leaves the tank at the same rate. Find an expression for the amount of salt in the tank at any time t and the limiting amount of salt in the tank as t rightarrow infinity. Find the time when the concentartion reaches 2 grams/liter.

Explanation / Answer

if p(t) is the amount of salt in the tank at any moment, then per minute ,addition of salt=4 gm/ltr * 2ltr/min=8 gm/min per minute,amounf of salt that leaves=(p(t)/120) gm/ltr * 2 ltr/min=(p(t)/60) gm/min so dp/dt=8 - p(t)/60 with p(0)=0 solving the differential equation,we get p(t)=480*(1-exp(-t/60)) so when t->infinity, we have amount of salt=480 grams b)concentration =2 gm/ltr total amount=120*2=240 grams so 480*(1-exp(-t/60))=240 t=41.58 sec

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