Question

Suppose that a large mixing tank initially holds 800 gallons of water in which 50 pounds of salt have been dissolved. another brine solution is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at a slower rate of 2 gal/min. if the concentration of the solution entering is 2 lb/gal, determine a differential equation for the amount of salt a(t) in the tank at time t > 0. (use a for a(t).)

Answer 1

The initial amount of salt in the tank is 50 pounds, so the initial condition for the solution is
a(0) = 50

The rate of influx of salt is (3 gal/min)*(2 lb/gal) = 6 lb/min.

The rate of outflow of salt is "a" times the proportion of tank contents being pumped out in a minute. The tank contains 800 gallons initially, and the quantity is increasing at (3-2)=1 gallon per minute. Solution is being pumped out at 2 gal/min. Thus the fraction of salt being removed is 2/(800+t).

a' = 6 - (2/(800+t))a