Differential Equations Mixture problem

[Jeff12341234]

at t=o an evenly mixed 40 liters solution with 100grams of salt is poured into a container. solution comes in at 2 grams per liter at 3 liters per minute and flows out at the same rate. At what time is the salt solution 120 grams?

I got -9.something minutes for my answer :/

This question was on a test I just took. It's from memory. The linear equation I got was
dA/dt+3/40*A=6

What should the answer be?

 

[haruspex]

Your equation looks right.
What is the initial concentration? What is the target concentration? What is the concentration coming in? Any surprise here?

 

[rude man]

at t=o an evenly mixed 40 liters solution with 100grams of salt is poured into a container. solution comes in at 2 grams per liter at 3 liters per minute and flows out at the same rate. At what time is the salt solution 120 grams?

I got -9.something minutes for my answer :/

This question was on a test I just took. It's from memory. The linear equation I got was
dA/dt-3/40*A=6

What should the answer be?

There is 6g of salt going into the vessel and 3x/40 g leaving, so the salt contents in the vessel is changing at a rate of
dx/dt = 6 - 3x/40
which is not your equation. Your equation is adding the stuff that is supposed to be leaving!

(Jeez, I hope I got that right ... will trade answers when appropriate).

 

[haruspex]

dx/dt = 6 - 3x/40
which is not your equation.

Good catch - I missed that. But I suspect it's just a transcription error in the post. The D.E as posted would not have led to a negative answer.

 

[Jeff12341234]

yea. I mistyped it. should've been dA/dt+3/40*A=6

The problem still remains though. He states that the mixture was just poured in at t=0 implying that a negative time is even more likely to be a wrong answer :/

 

[haruspex]

He states that the mixture was just poured in at t=0 implying that a negative time is even more likely to be a wrong answer :/

Pls try to answer my questions in post #2.

 

[Jeff12341234]

Your equation looks right.
What is the initial concentration? What is the target concentration? What is the concentration coming in? Any surprise here?

100 grams per 40 liters initial
120 grams target
2 grams per liter at 3 liters per minute is coming in

None of that helps me in any way :/

Is my answer correct?

 

[haruspex]

100 grams per 40 liters initial
120 grams target
2 grams per liter at 3 liters per minute is coming in

None of that helps me in any way :/

Is my answer correct?

Yes, but let's put that in consistent units... 2.5g/l initially; 2g/l coming in; 3g/l target. See a problem with that?

 

[Jeff12341234]

there's only a problem because the same amount that is coming in is also leaving correct? that's why it never gets to 120? to add to that, the resulting function showed the amount gradually decreasing over time.

 

[haruspex]

there's only a problem because the same amount that is coming in is also leaving correct? that's why it never gets to 120? to add to that, the resulting function showed the amount gradually decreasing over time.

Having expressed it in terms of concentrations, it doesn't matter whether the volume in the tank is increasing or decreasing. If you start with a solution more dilute than the target, and you add solution more dilute than the target, how are you ever going to reach the target concentration?

 

[Jeff12341234]

ok. i see

 

[rude man]

there's only a problem because the same amount that is coming in is also leaving correct? that's why it never gets to 120? to add to that, the resulting function showed the amount gradually decreasing over time.

It does get to 120. In 63.83 sec.

 

[Jeff12341234]

what function did you get?

 

[rude man]

what function did you get?

EDIT: right answer after all. Sorry to confuse.

OK, we all agree on the ODE by now I take it, so have you tried to solve it? Show your effort.

Sorry again, I see you did, but I can't make out the top part.

 

[Jeff12341234]

I edited the post above to show most of the work. You can't have negative time though especially since he says "at t=0 the well mixed solution was poured in" meaning that before that, the tank was empty

 

[rude man]

I edited the post above to show most of the work. You can't have negative time though especially since he says "at t=0 the well mixed solution was poured in" meaning that before that, the tank was empty

Hold everything, I have to reexamine what I did.

EDIT: the problem has no solution and haruspex is right.

 

[haruspex]

Looking back on this problem, you might consider that the problem setter made an error. OTOH, it's quite a good exercise in recognising the significance of an apparently silly answer.