# Differential Equation Mixing Problem

## Homework Statement

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.

## 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.

## Answers and Replies

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

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