Tank concentration-of-chlorine problem (Dif. Eq.)

[StrappingYL]

Homework Statement

A tank with a capacity of 400 L is full of a mixture of water and chlorine with a concentration of 0.05 g of chlorine/liter. The chlorine concentration is to be reduced by pumping in fresh water at the rate of 4 liters/second. The mixture is kept stirred and pumped out a rate of 10 liters/second. What is the concentration of the chlorine in the tank 15 seconds later?

The Attempt at a Solution

Set x = the amount of chlorine in the main tank.

Well there isn't any chlorine entering the main tank, so the mixture going in would be 4, and the amount going out would be 4*(x / 400). So my setup is:

(dy/dt) = 4 - 4*(x / 400)

And I don't really know where to go from there. I don't even know if my setup is correct. I'm still trying to get the intuition of differential equations, lol.

Thanks!

 

[Dick]

That isn't a very good start. You haven't even said what 'y' is supposed to be. Here's a hint. Let V be the volume of solution contained in the tank. Can you express V as a function of t? Now can you express dx/dt in terms of V and x?

 

[StrappingYL]

That isn't a very good start. You haven't even said what 'y' is supposed to be. Here's a hint. Let V be the volume of solution contained in the tank. Can you express V as a function of t? Now can you express dx/dt in terms of V and x?

I'm sorry, I meant:

dx/dt = 4 - 4*(x / 400)

Sorry for my mistake. For my volume function:

V(t) = 400x + 4 - 10t(x / 400)
dV/dt = 4 - x / 100

Am I on the right track?

EDIT: And x = 5 / 100 when t = 0, correct?

 

[Dick]

I'm sorry, I meant:

dx/dt = 4 - 4*(x / 400)

Sorry for my mistake. For my volume function:

V(t) = 400x + 4 - 10t(x / 400)
dV/dt = 4 - x / 100

Am I on the right track?

Mmm. Not really. I think you can express V(t) without any reference to x. Can't you? Just concentrate on that for now.

 

[StrappingYL]

V(t) = 0.05 - (0.05 / 400)*6*t ?

I'm sorry, I'm still trying to get the intuition of differential equations. I know the result will pretty much always be a function, but it's difficult for me to comprehend what a differential is equal to. I know an equation contains two of the same quantities, but an equation of differentials is outside my usual scope of comprehension. So I don't really know how to set it up. My professor is at a science convention in D.C. for a few days and I'm really trying to learn these without his guidance, lol. I hope this forum's denizens are the patient types :)

EDIT: 6 not 10 because of the 4 liters going in, right?

 

[Dick]

I'm patient. But I don't stay up all night either. Look V(0)=400 L, right? And then, yes, V(t) goes down by 6 L per second. I think you are overcomplicating this. Tell me quick. What's V(t)?

 

[StrappingYL]

I'm sorry, nevermind I never figured it out. Thank you for your help.