# Differential Equation Mixing Problem

1. Sep 4, 2012

### cowmoo32

1. The problem statement, all variables and given/known data
Suppose the clean water of a stream flows into Lake Alpha, then into Lake Beta, and then further downstream. The in and out flow for each lake is 200 liters per hour. Lake Alpha contains 500 thousand liters of water, and Lake Beta contains 400 thousand liters of water. A truck with 200 kilograms of Kool-Aid drink mix crashes into Lake Alpha. Assume that the water is being continually mixed perfectly by the stream.

Let x be the amount of Kool-Aid, in kilograms, in Lake Alpha t hours after the crash. Find a formula for the rate of change in the amount of Kool-Aid, dx/dt, in terms of the amount of Kool-Aid in the lake x.

3. The attempt at a solution
$\frac{dx}{dt}$=xin-xout

I was thinking it was:

$\frac{200L}{hr}$-($\frac{200x kg}{500000L}$x$\frac{200L}{hr}$)

=200-.08x

But I've gone wrong somewhere.

2. Sep 5, 2012

### voko

If you have x Kool Aid at time t, how much of it will go away in delta t?

3. Sep 5, 2012

### HallsofIvy

Staff Emeritus
If X(t) is the amount of Kool-ade in lake alpha, how much Kool-Ade is there in each liter of water? Since water is flowing out of lake alpha at 200 liters per hour, how much Kool-ade is taken out of lake alpha every hour? There is NO Kool-ade coming in. And lake beta is irrelevant to this problem.