Thread: Salt water pumped into a tank problem.

Suppose that a large mixing tank initially holds 400 gallons of water in which 65 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 6 gal/min, and when the solution is well stirred, it is pumped out at a faster rate of 8 gal/min. If the concentration of the solution entering is 3 lb/gal, determine a differential equation for the amount of salt in the tank at time

Hey derail.

What is the rate of change of salt? Is the density of salt the same for all forms of brine?

Since the tank initially "holds 400 gallons of water" and "brine solution is pumped into the tank at a rate of 6 gal/min" and "it is pumped out at a faster rate of 8 gal/min" so after t minutes it will hold 400+ 6t- 8t= 400- 2t gallons of water. The density of the brine is the amount of salt, A(t), divided by the volume of water, 400- 2t, is . The point of that is that 8 gallons of water being pumped out each minute will contain pounds of salt. At the same time, a brine solution with 3 lbs/gal is coming in at 6 gal/min and so salt is coming in at (3 lbs/gal)(6 gal/min)= 18 gal/min.

dA/dt is the rate at which salt is coming into the tank (positive for inflow, negative for outflow) so .