Differential Equation

Question

A 120-gallon tank initially contains 80 lbs. of salt dissolve in 90 gal of water. Brine

containing 2lbs/gal of salt flows into the tank at the rate of 5 gal/min, and the mixture

flows out of the tank at the rate of 4 gal/min. If the tank is full after 40 minutes, how

much salt does the tank contain?

Answer 1

Let S(t) = lbs of salt in the tank at time t. Based on the information given, I obtain the differential equation

dS(t)/dt + {4/(90 + t)}S(t) = 10 being a first order DE, you can use the standard integrating factor to obtain

∫ d(S(t)exp(4Ln(90+t)) = ∫ 10exp(4Ln(90+t) dt

If you integrate from t=0 to t=30 minutes you obtain 183.05 lbs of salt

30 minutes is when the tank is full since it rises 1 gal/min giving 30 minutes to full

However, the problem asks for the salt at 40 minutes

That result is 218.65 lbs of salt if the tank could hold it?