Initially 5 grams of salt are dissolved in 20 liters of water. brine with concentration of salt 2 grams of salt per liter is added at a rate of 3 liters a minute. the tank is mixed well and is drained at 3 liters a minute. how long does the process have to continue until there are 20 grams of salt in the tank?

3.731 minutes

Explanation:

Let the amount of salt in the tank at any time be x(t)

Since x(0)=5 g is  dissolved in 20 liters of water

Brine with 2 grams per liter salt enters the tank at the rate of 3 liters/min

Salt entering per minute is 2* 3=6 grams/min

Volume of liquid leaving the tank is the same as the volume of liquid of tank entering, 3 liters/min

volume of liquid remains at 20 liters at all times

At any given points of time, the concentration of salt is grams/liter

Amount of liquid leaving per minute is 3 liters/min so that the amount of salt leaving is grams/minute

Differential equation governing the salt amount in the tank is

Therefore,    

Integrating factor is   and so the equation becomes

Therefore,  

Using the initial condition  

  is the amount of salt at any point of time

After approximately 3.731 minutes, we have 20 grams of salt in the tank