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I was given the task of writing a formula for the following scenario:

A tank with a capacity of 3,000 cubic meters is filled with a substance. We want that substance, we'll call it substance1, to be diluted by substance2 until substance1 equals 1% of its original volume. Substance2 will be introduced to the tank by filling up a balloon, and then popping it, allowing the substances to mix. Before the balloon and substance2 are introduced to the tank, we need to empty an equal amount from what's already in the tank.

The dilution will occur in steps:

  1. A balloon will be filled with X-Cubic-Meters of substance2.

  2. X-Cubic-Meters will be released from the tank.

  3. The balloon will be put into the tank and popped, allowing the substances to mix thoroughly.

Steps 1 - 3 will repeat until substance1 reaches 1% of its original amount. I believe the problem relates to exponential decay due to how substance1 is both released from the tank, and diluted in each iteration of the process.

Squanch
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  • This looks more like a discrete problem to me because we are changing the concentration by fixed volumes in discrete steps. Exponential decay applies to continuous changes. – John Douma Nov 04 '18 at 05:54
  • I think I'm following what you're saying. But in addition to the amount of substance2 we pump in, we release some of the mixture in the tank prior to letting substance2 in. So the amount of substance1 getting released from the tank gets less and less with each step, since the substances mix together after each iteration. Man, this is hard to explain. Sorry. – Squanch Nov 04 '18 at 18:09
  • That's right. Imagine you have a storehouse full of gold ingots. Each night I replace one with an ingot of fools gold. Isn't that the same thing? – John Douma Nov 04 '18 at 18:17
  • Close, but before you add the fools gold, you have to release an amount of gold from the storehouse to accommodate the fools gold. and after the fools gold is added, what's in the storehouse is melted together completely. So the next time you want to release gold to accommodate the fools gold, the mixture being released isn't 100% gold. Does that make sense? I think we figured out an equation, lemme figure out how to format it. – Squanch Nov 04 '18 at 19:28

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