When is the tank empty? [differential equation]

[Rijad Hadzic]

Homework Statement

problem 23:

A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 lbs of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well mixed solution is pumped out at the same rate. Find the number A(t) of lbs of salt in the tank at time t.

problem 25: Solve problem 23 under the assumption that the solution is pumped out at a faster rate of 10 gal/min. When is the tank empty?

I'm only doing problem 25 here. I've already done problem 23.

Homework Equations

A(0) = 500

The Attempt at a Solution

I really don't know where to start. Discussing with someone, they said that the "brine" is important for this problem, but I really do not understand how.

It is asking for when the tank is empty. I don't see how the concentration of salt has any importance here. I don't see how the amount of salt that is present in a diluted amount of solution changes how much solution there is.

I have $\frac {dA}{dt} = R_{in} - R_{out}$

So at time t=0, 500 gallons of water are present

$5 \frac {gal}{min}$ of water, saturated heavily with salt at 2lbs per gallon is being pumped in but this isn't even relevant to this problem because its only asking about when water in the tank is 0, not about a concentration.

After t minutes, since the incoming amount of water is 5 gallons/min, and the water coming out is 10 gallons/min, you have -5 gallons per minute coming out.

At this point I'm lost. Since I'm trying to make an equation here, and 5 gallons/min is already in units, if I try to add a variable t like 5t, it makes the units just 5 gallons, so that's no good.

Can someone please not spoil the problem, but just confirm to me that the concentration of salt really doesn't matter? Or if it does, can anyone explain why, because I really don't see it right now. In the mean time I'm going to try to figure out how I can make an equation out of this.

 

[Ray Vickson]

Homework Statement

problem 23:

A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 lbs of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well mixed solution is pumped out at the same rate. Find the number A(t) of lbs of salt in the tank at time t.

problem 25: Solve problem 23 under the assumption that the solution is pumped out at a faster rate of 10 gal/min. When is the tank empty?

I'm only doing problem 25 here. I've already done problem 23.

Homework Equations

A(0) = 500

The Attempt at a Solution

I really don't know where to start. Discussing with someone, they said that the "brine" is important for this problem, but I really do not understand how.

It is asking for when the tank is empty. I don't see how the concentration of salt has any importance here. I don't see how the amount of salt that is present in a diluted amount of solution changes how much solution there is.

I have $\frac {dA}{dt} = R_{in} - R_{out}$

So at time t=0, 500 gallons of water are present

$5 \frac {gal}{min}$ of water, saturated heavily with salt at 2lbs per gallon is being pumped in but this isn't even relevant to this problem because its only asking about when water in the tank is 0, not about a concentration.

After t minutes, since the incoming amount of water is 5 gallons/min, and the water coming out is 10 gallons/min, you have -5 gallons per minute coming out.

At this point I'm lost. Since I'm trying to make an equation here, and 5 gallons/min is already in units, if I try to add a variable t like 5t, it makes the units just 5 gallons, so that's no good.

Can someone please not spoil the problem, but just confirm to me that the concentration of salt really doesn't matter? Or if it does, can anyone explain why, because I really don't see it right now. In the mean time I'm going to try to figure out how I can make an equation out of this.

It looks to me as though problem 25 has two parts. Part (i): what is salt content A(t) as a function of t?; and (ii) when is the tank empty?

Certainly, (ii) has nothing to do with (i), but if you read what is actually written it said to solve problem 23 in the new setup, and that problem asked for A(t).

 

[Rijad Hadzic]

It looks to me as though problem 25 has two parts. Part (i): what is salt content A(t) as a function of t?; and (ii) when is the tank empty?

Certainly, (ii) has nothing to do with (i), but if you read what is actually written it said to solve problem 23 in the new setup, and that problem asked for A(t).

Gosh you are a life saver. I don't know how I missed the "solve problem 23" part even though I clearly wrote it out.

Will report back in a minute.

 

[Rijad Hadzic]

I got the answer, I'm marking thread as solved for now.