1. ## Mixing problem

A 200-gallon tank is half full of distilled water . At time t=0, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 5 gallons per minutes, and the well-stirred mixture is withdrawn at the rate of 3 gallon per minutes.

At the time the tank is full (t=50mins), how many pounds of concentrate will it contain?

The ans is 82.32lb. can anyone tell me how to do it?

2. Originally Posted by AAA
A 200-gallon tank is half full of distilled water . At time t=0, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 5 gallons per minutes, and the well-stirred mixture is withdrawn at the rate of 3 gallon per minutes.

At the time the tank is full (t=50mins), how many pounds of concentrate will it contain?

The ans is 82.32lb. can anyone tell me how to do it?
Well, I presume you understand this is a differential equation problem. (I assume this is for a differential equations course.)

Let X(t) be the pounds of concetrate in the tank at time t minutes. Since there are 5 gallons of water coming into the tank per minute and 3 gallons going out each minute, their are a net 5- 3= 2 gallons coming in each minute and the volume of water in the tank is 100+ 2t at time t minutes (so it will be full when 100+2t= 200, 2t= 100, t= 50 min as given). The concentration at time t min. is X(t)/(100+ 2t).

dX/dt is the net rate at which concetrate is going into the tank. We know that the 5 gallons of water coming into the tank each minute carries in 0.5 pounds per gallon. How many pounds is that per minute coming in?

Water goes out at 3 gallons per minute and carries X(t)/(100+2t) pounds with it. How many pounds is that per minute?

What is the net rate at which the concentrate is coming in per minute? Can you set up the differential equation from that?