A tank of blood which is 50 gallons in volume is full of this red delicious substance. The man who owns this tanks want to replace the blood with ethanol. Since the tank is heavy he cannot pour all the water out. Rather he has a 1 gallon pail. Each time he pours the ethanol in the mixture displaces 1 gallon.
He had 50 gallons of blood.
He pours 1 gallon of ethanol and the blood displaces 1 gallon.
Now he has 49 gallons of blood and 1 gallon of ethanol.
He pours 1 gallon of ethanol now 1 gallon of the mixture displaces (not the blood, otherwise this be too easy).
And so on.....
1)Show that the man can never fully (ideally) clean the tank to ehtanol only.
2)But he would be satisfied with if the concetration of blood in the tank is at least .01%
How many pouring are required?
This problem seems awesomely similar to your differencial equations rate in/rate out mixture equation. But it is not. Because this is not a continous time time. The amout of concetration is not based on the time and continuity of it going in and out but rather on the number of pouring which is a natural number and hence a discrete model.
With each pouring, 2% of the mixture is displaced.
Hence, 98% of the blood remains.
After n pourings, there are: B = 50(0.98)^n gallons of blood in the tank.
We can see that B will never equal 0 (in a finite number of pourings).
If B < 0.01%, we have: 50(0.98)^n < 0.0001 . . . at most 0.01%
Then: n = ln(0.000002) ÷ ln(0.98) = 649.5348951
Therefore, it will take 650 pourings.
This was the blood of a murdered Soviet whose blood was full of ethanol.
I will solve this problem in a general sense.
Let be the volume in whatever units.
Let be the volume of the pail.
Define a sequence,
as the amount of ethanol in the tank.
The initial condition is,
To solve this "differencial equation look-a-like" we will define a recurrence relation. To define through .
(because of displacement)
because you pour in a constant amount of ethanol.
because you have amount of ethanol and because it is a homogenous solution everything is divided evenly. Thus the amount displace is the one shown above.
The pattern continues...
It takes a trained eye to recognize the geometric series.
Thus, we have,
The rest is trivial all you need to solve where is the amount specified in the problem.
The problem is not mine while the solution is. I took this problem from the Talmud which deals with rainwater. I did not check Soroban's solution but it is probably correct and shorter than mine.