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Salty Water – Fall 2014

October 27, 2014

The question. A big storage tank, partially full of slightly salty water, is perpetually leaking into the ground at the rate of 10,000 gallons per day. To compensate for this pure water is steadily added to the tank at the rate of 10,000 gallons per day. An inspection found that the tank contains 10,000 gallons and 1 lb of salt; the inspection also found that the conditions in the tank are keeping the tank thoroughly stirred. How much salt is there in the tank a day after the inspection?

Ideas from the first week:

One thing that everyone agreed on was:  “less salt is leaving each time”

The other main conjectures are that there will be some very small amount of salt left after one day, or that there will be 1/2 lb of salt left after one day.

One idea is to try to simplify the problem.  One possible simplification is to try working with 1 gallon of water, instead of 10,000 which is cumbersome.

Ideas from the second week:

Pretend that the day passed all at once in 1/2 day chunks, then 1/3 day chunks, etc.

A formula for how that happened in a general case was offered. The joke was that it was a very “simple” formula but it was actually quite complicated.

Ideas from the third week:,

The formula from last week formula was transformed into : [(n-1)/n]^n

We worked for a while trying to connect this to last week’s formula.

Then,  [(n-1)/n]^n was simplified to: [1 – 1/n]^n and there was a definite aha! moment when others saw that this meant 1/n amount of salt left in each 1/n chunk of time.

Someone also noted that when you set n = 10,000,000 then the formula equals 0.367 which was thought to be 1/e. Nobody really knew what e was.

At the very end, we started looking at the second problem, with 50,000 gallons……….

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