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 :tongue2: ... 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.