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 100 liters per hour. Lake Alpha contains 400 thousand liters of water, and Lake Beta contains 100 thousand liters of water. A truck with 500 kilograms of Kool-Aid drink mix crashes into Lake Alpha. Assume that the water is being continually mixed perfectly by the stream.

1. 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 $x$.
   $\dfrac{dx}{dt} =$ kg/hour
   
   
2. Find a formula for the amount of Kool-Aid, in kilograms, in Lake Alpha $t$ hours after the crash.
   $x(t) =$ kg
   
   
3. Let $y$ be the amount of Kool-Aid, in kilograms, in Lake Beta $t$ hours after the crash. Find a formula for the rate of change in the amount of Kool-Aid, $dy/dt$, in terms of the amounts $x, y$.
   $\dfrac{dy}{dt} =$ kg/hour
   
   
4. Find a formula for the amount of Kool-Aid in Lake Beta $t$ hours after the crash.
   $y(t) =$ kg