Question

A tank initially contains 400 gallons of fresh water. Water containing 0.2 lb of salt per gallon enters the tank at a rate of 1 gallon per minute, and the well-mixed solution is drained from the tank at a rate of 2 gallons per minute. If () represents the amount of salt in the tank at time , what is the differential equation governing the rate of change of ()?

Answer 1

The differential equation governing the rate of change of salt () in the tank is d()/dt = 0.2 - 2 * ().

Step-by-step explanation:

The differential equation governing the rate of change of salt () in the tank can be determined using the principle of mixing. Let t represent time in minutes. The rate of change of () is equal to the rate of salt entering the tank minus the rate of salt being drained from the tank. Since water containing 0.2 lb of salt per gallon enters at a rate of 1 gallon per minute, the rate of salt entering the tank is 0.2 lb/min. On the other hand, the well-mixed solution is drained from the tank at a rate of 2 gallons per minute, so the rate of salt being drained is 2 * () lb/min. Therefore, the differential equation is given by d()/dt = 0.2 - 2 * ().