Lisa: I don't think they're giving enough information, Dad.
Homer: I'll figure it out. I'm gonna use all the power of my brain.
Sometimes you just don't have enough information, even if you use all the power of your brain.
Today I vaguely considered whether I wanted to dye another batch of yarn using some green and yellow dye I have (I decided not to), and I remembered that the last time I'd used that yellow, it was all absorbed almost instantly into the yarn, leaving the dye bath what they call "exhausted". And then naturally I tried to figure out where "exhausted" might have come from. I could see that all the common uses of the word had overlapping meanings: a dye bath can be exhausted and so can a person, meaning they have nothing left in them, and the exhaust of a vehicle is the remnants — what's thrown away after the useful portion of it has been utilized.
But as it turned out, I didn't have enough information, because the root of the word (obviously Latin) is so rare in English that it has no other common relatives, with, as far as I know, only one other tiny group of words employing it.
That root is "haurire", "to draw up", which specifically referred to water in Latin. The addition of the "ex-", "off", to the beginning gives it the sense of completion: all the water has been drawn off or out of something, leaving it empty — exhausted, which is what a person is when they have no energy left in them (or a dye bath when it's out of dye molecules).
The other little batch of English words is from the field of biology, and includes such useful terms as "haustorium", a tube which a fungus uses to siphon up nutrients, and "Haustrum", a genus of sea snail that drills through its prey's shell with a feeding tube.