Your swimming pool containing 60,000 gal of water has been contaminated by 6 kg of a non toxic dye that leaves a swimmer's skin an unattractive green. the pool filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of 200 gal/min. (a) what is the initial value problem for the filtering process; let q(t) be the amount of dye in the pool at any time t. what is the initial amount of dye in the pool?

You can treat this is like a mixing problem where pure water (not containing the dye) is pumped into a tank that hold 60,000 gal of water. The rates at which the amount of dye is flowing in and out of the pool are, respectively,

Then the net rate at which the amount of dye in the pool changes is such that

Initially, the pool contains of dye.

The ODE is separable as

Given that , you get , so the amount of dye in the pool at any time is governed by

constant of proportionality = 3 and the equation is c = 3p.

step-by-step explanation:

c = kp   where c = amount of caffeine in mg., p = oz of pepsi and k is the constant of proportionality.

substituting the values given:

24 = 8k

k = 3.

the same value of k will be obtained if we substitute the other given values.