I have a fluid mixing problem with my car and I can't seem to find the answer: I have to change my transmission oil. The transmission has a 7 liter total capacity, but due to the torque converter, I can only drain 4 liters at a time. Thus, I have to make several partial changes. The question is:
- How many partial changes do I have to do in order to have a (almost) clean transmission fluid (< 5% old)?
Additionally:
- What area of Math is that problem?
- Is there a theorem or formula that explains that specific problem?
Every time you change the transmission fluid, you are multiplying the ratio of old fluid to total fluid by $3/7$. So after the first change, you have a $3/7\approx 42\%$ old fluid, after two changes you have $9/49\approx 18\%$ old fluid, etc... The general expression is that the ratio of old fluid is $$r=(3/7)^n$$ where $n$ is the number of changes. Given that you want $r\leq .05$, we can solve this using logarithms: $$n\geq\frac{\log(.05)}{\log(3/7)}\approx 3.5.$$ So, you would have to do about three and one-half fluid changes to make sure that you have sufficiently clean transmission fluid. Of course, this method assumes that at the start of each change, the new and old fluid is mixed homogeneously, so make sure you run the engine a bit between each change.
The math I used here is algebra, but a related situation (which may not be possible with a real car) is if you are continuously draining fluid from the transmission and at the same time replacing it with new fluid. The amount of old fluid in the transmission at any given time is a smooth function of time which depends on the rate of draining and refilling. This is a standard question in introductory differential equations.