We begin with a tank containing 100 gallons of water and 50 pounds of dissolved salt. Pure water is added to the tank at a rate of 3 gallons per minute (the concentration of the salt is kept constant by stirring). If the mixture runs out of the tank at 2 gallons per minute, how much salt (in pounds) is in the tank at the end of 60 minutes?
Let be the number of pounds of salt in the tank at time . Since water leaving the tank at a rate of 2 gallons per minute, and at time there are gallons of water in the tank (since water is entering the tank at a rate of 3 gallons per minute) we have
(Since is the change in the amount of salt at time and is the concentration of salt in the water at time .) Therefore, we have the first-order differential equation
Applying Theorem 8.3 (page 310 of Apostol) with
Therefore, at time we have