Suppose the clean water of a stream flows into Lake Alpha th

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 100 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 incremental change in the amount of Kool-Aid, ?x , in terms of the amount of Kool-Aid in the lake x and the incremental change in time ?t . Enter ?t as Deltat.
?x= kg

Find a formula for the amount of Kool-Aid, in kilograms, in Lake Alpha t hours after the crash.
x(t)= kg

Let y be the amount of Kool-Aid, in kilograms, in Lake Beta t hours after the crash. Find a formula for the incremental change in the amount of Kool-Aid, ?y , in terms of the amounts x,y , and the incremental change in time ?t . Enter ?t as Deltat.
?y= kg

Find a formula for the amount of Kool-Aid in Lake Beta t hours after the crash.
y(t)= kg

Solution

A>

There is no additional Kool-Aid coming into Lake Alpha once the truck has crashed and deposited the 200 kg there, so the IN rate is zero.

THe out flow rate would be

(x/100000)*500 = x/200

And so this gives us the IVP:

dx/dt = -x/200 , x(0) = 200

B>

For this part we\'ll have to solve

dx/dt = -x/200 , x(0) = 200

x(t) = 200e^(-t/800)