Understanding Mixture Problems: Solving for Tank Overflow Time

[prace]

Hello,

I think this is a pretty easy question here that I think I may be just overlooking. I am asked this huge drawn out DE problem about a tank being filled with water and brine and I am asked to solve all these different parts. My problem is that the first part of the question asks to find out when the tank overfills, and I can't seem to get that. I know it is just simple formula, but for some reason I am just not seeing how it is formed. Word problems definitely are not my strong point. If anyone can explain the thinking behind this, it would be a great help. Thanks.

Here is the problem:

A 400 gallon tank is filled with 300 gallons of fluid in which 50lbs of salt is dissolved. Brine containing 2 pounds of salt per gallon is pumped in at a rate of 3 gallons per minute. The well-mixed solution is pumped out at a rate of 2 gallons per min. When will the tank overflow?

I was given the answer in class by my professor to be 100 minutes and the formula was 300+2t = 500, but I have no idea where this came from.

There is a lot more to the problem, but that is the first part and I need to get past that to get to the rest.

 

[Hootenanny]

We know that initially, that the tank contain 300 gallons of fluid and its capacity it 400 gallons. If we consider the net flux (\Phi) into the tank then we obtain;

300+\Phi=400

Now, the net flux will be the flux of liquid into the tank minus the flux of water out of the tank. Can you go from here?

 

[prace]

Sorry ... I am pretty lost here. I have a basic understanding of what flux is, but I am not 100% clear. Let me see if I can understand this correctly...

It seems to me that (\Phi) would be the flux of liquid into the tank (3 gal/min) minus the flux of liquid out of the tank (2 gal/min) , or 3-2=1. 300+t=400, then t=100. So that means that the tank will overflow in 100 minutes. Wow, that just sort of came out while I was typing here. haha. Makes sense. So then, how come the formula given is 300+2t=500, and not 300+t=400?

 

[prace]

As I look at this equation a little closer, I see that my last post is not entirely correct. I said that

the flux of liquid into the tank (3 gal/min) minus the flux of liquid out of the tank (2 gal/min) , or 3-2=1. 300+t=400, then t=100.

, but in the original problem, it says that the salt solution is being pumped in at a rate of 3 gal/min and being pumped out at 2 gal/min. That being said, it also means that the the rate of the overflow of the tank will be 1 gal/min. So to me, that means that the total flux of solution going out of the tank would be 2 gal/min + 1 gal/min = 3 gal/min. Then that makes the previous equation of 300+t=400 not true since t = 0, and for sure 300 does not equal 400 ... So back to square one for me.

So, here is my thinking. Is rate out (when overflowing) not the total rate out (the amount being pumped out + the amount overflowing), but just the amount overflowing? That would make the formula given to me by my professor make sense with the 300+2t = 500, except, I don't know where that 500 came from and that thinking is a little counter intuitive for me. Am I totally going off in the wrong direction here?

 

[Hootenanny]

The reason that your equation does not 'work' at t equal to zero is that we want to find the time when it overflows, that is our condition. Look at it this way, the volume V of solution in the tank at time t will be given by;

V=300+t

Now, we want to know when the tank will overflow, i.e. when V>400, therefore we have;

V>400

V=300+t

Substituting the second equation into the inequality we obtain;

300+t>400

Is that more satisfactory?

 

[prace]

ok, I get that, but how does that help me here? I mean, yes, the tank will overflow at some time t and I know that at that time t it will have to be more than 400 gallons, or V > 400, but I still do not see what that does for me here except help me visualize what is going on.

 

[prace]

Any help on this one anyone?

 

[Hootenanny]

ok, I get that, but how does that help me here? I mean, yes, the tank will overflow at some time t and I know that at that time t it will have to be more than 400 gallons, or V > 400, but I still do not see what that does for me here except help me visualize what is going on.

I don't follow what your problem is? Surely you know all you need to do now is solve for t?

 

[prace]

Yes, I solve for t and I get 100 minutes, which is the correct answer, however, I do not see how the formula given in my class (300+2t = 500)makes sense. I am sorry to make you spell this out for me and go through all this, but I really appreciate you helping me. Thank you!

 

[Hootenanny]

Yes, I solve for t and I get 100 minutes, which is the correct answer, however, I do not see how the formula given in my class (300+2t = 500)makes sense. I am sorry to make you spell this out for me and go through all this, but I really appreciate you helping me. Thank you!

I'm not sure how your Prof derived the equality but you can reach the one I derived from your Prof's by taking your equation, dividing throughout by two and adding 150 units to each side, but I have no idea why your Prof would want the equality in this form unless it is relevant for further questions.