# Mixing Tank, Differential Equations Problem

1. Feb 21, 2012

### bdh2991

1. The problem statement, all variables and given/known data

a larg tank is filled with 500 gals of water with 400lbs of salt. pure water is pumped into the tank at a rate of 3gal/min. the well mixed solution is pumped out at a rate of 7 gal/min.

I need help finding the Differential

2. Relevant equations

3. The attempt at a solution

I tried coming up with the D.E. and this is what i got dA/dt = 3 - 7A/(500-t)

except after i did solved the d.e. i kept getting more salt than i started with which makes no sense...

2. Feb 21, 2012

### fauboca

Your denominator is wrong. How is rate out defined?

3. Feb 21, 2012

### bdh2991

not sure lol...i went off of a similar problem that was gaining instead of losing...

i know that it is supposed to be the flow rate out multiplied by the concentration out but i don't understand how to get the concentration of salt exiting the tank...

4. Feb 21, 2012

### fauboca

$$R_{out} = -t(7gal/min - 3gal/min) = -4t$$

Where you have 500-t. I didn't check over the rest of you equation but that was what was immediate obvious.

5. Feb 21, 2012

### bdh2991

ok so instead i should have A/(-4t) rather than A/(500-t)?

6. Feb 21, 2012

### fauboca

No, why would you disregard the amount of solution?

7. Feb 21, 2012

### bdh2991

i guess i'm misunderstanding what you are saying....it would help more if you just showed me what the equation should look like

8. Feb 21, 2012

### fauboca

I could have phrased it better.

When you are dealing with mixtures, you denominators is

$$\frac{A}{sol+(R_{in}-R_{out})t}$$

When inflow and out flow are the same, we just have A/sol in gallons.

Your R_{in}=3 and out=7 so (3-7)t = -4t.

So what is the denominator?

9. Feb 21, 2012

### bdh2991

A/500-(-4t)

so A/500+4t?

10. Feb 21, 2012

### bdh2991

it can't be that because i just tried it and it didn't come out right

11. Feb 21, 2012

### fauboca

500+(-4t) = 500-4t

12. Feb 21, 2012

### bdh2991

ok i made some dumb mistakes but i finally got what i needed thanks!