Exercise 9

Suppose that you have a water tank that holds 100 gallons of water. A briny solution, which contains 20 grams of salt per gallon, enters the tank at the rate of 3 gallons per minute.

At the same time, the solution is well mixed, and water is pumped out of the tank at the rate of 3 gallons per minute.

1. Since 3 gallons enter the tank every minute and 3 gallons leave every minute, what can you conclude about the volume of water in the tank?

2. How many grams of salt enter the tank every minute?

3. Suppose that $S(t)$ denotes the number of grams of salt in the tank in minute $t$. How many grams are there in each gallon in minute $t$?

4. Since water leaves the tank at 3 gallons per minute, how many grams of salt leave the tank each minute?

5. Write a differential equation that expresses the total rate of change of $S$.

6. Identify any equilibrium solutions and determine whether they are stable or unstable.

7. Suppose that there is initially no salt in the tank. Find the amount of salt $S(t)$ in minute $t$.

8. What happens to $S(t)$ after a very long time? Explain how you could have predicted this only knowing how much salt there is in each gallon of the briny solution that enters the tank.