# Classic watertank equation

1. Jun 4, 2008

### cpx

[SOLVED] Classic watertank equation

I'm having trouble with a variant of the classic watertank equation. The data is as follows.
A tank contains 300 liters of saline water, containing a total of 1800 grams of salt. Through an inlet, saline water containing 5 grams/liter is pumped in at a speed of 2 liters/minute. The well-mixed solution is pumped out at a speed of 3 liters/minute. Compute the quantity of salt, in grams, after 100 minutes.

Here's my attempt at solving this:
$$V(t)=300-t$$

$$\frac{dS}{dt}=10t-3\frac{S}{V}$$

$$S(0)=1800$$

Running it in the ODE Analyzer in MAPLE got me $$S(100)\approx33867$$, which isn't the solution. Can anyone spot what I've done wrong?

2. Jun 4, 2008

### Dick

I don't see anything wrong with the setup. On the other hand if I integrate it I don't get 33867. Check your MAPLE setup. What are you supposed to get?

3. Jun 4, 2008

### Vid

Is it supposed to be 1089? I think the rate in should be just 10 and not 10t.

4. Jun 4, 2008

### Dick

Ooops. That's correct. I missed that! No wonder I didn't get 33867.

5. Jun 4, 2008

### cpx

Ah! Of course it should be just 10. I'm also getting 1089 now.

Unfortunately though, this doesn't seem to be the correct answer either. I don't have access to the correct solution; the task is available in a web form and it only returns whether the solution is correct or not. So either there's something else we've all missed, or the task is misformulated or the stored solution incorrect

In any case, thanks for the help! :)

6. Jun 4, 2008

### Vid

Well, the exact solution is 1088.89 or at least that's what mathematica tells me. A lot of web based assignments are finicky about these things.

7. Jun 4, 2008

### Dick

I didn't use numerics and got 9800/9. Where you supposed to approximate it? You can do it exactly.

8. Jun 5, 2008

### cpx

Nope. It's supposed to be rounded to the nearest integer..

9. Jun 5, 2008

### HallsofIvy

Staff Emeritus
Through the inlet, 10 grams of salt is coming in each minute. That first term should be "10" not "10t".

10. Jun 9, 2008

### cpx

It turned out the web assignment had the wrong answer stored. It's fixed now and the solution $$S(100)\approx1089$$ is correct. Thanks for the help! :)