Puzzle, possibly easy, mathematical.

[No Skid Marks]

So I have a 600litre fish tank, I usually change 1/3rd of the water by removing 1/3rd and replacing it with fresh water. If I was to do it by removing and replacing 10 liters at a time, how many 10 liters would it take to Acheive the same water quality? So if the fresh water was blue dye how many 10 liter changes would get the same shade of blue as if I had replaced 1/3rd(200 litres) in one go?
the details are fictional for arguments sake, I'm not asking for fish tank maintanence advice, just an answer to the mathematical equation asked.

 

[c3024446]

25 times...

 

[moorey]

This sounds like homeopathy.

 

[c3024446]

All in one go – take 200L out, replace with 200L of fresh water, so 400L of dirty water.

10L increments:

After the first one you have taken out 10/600 = 1.6667% of the dirty water, so there is 590L of dirty still in there
After the second one, taking another 1.66667% out, so now 580.16667L
…..
After the 25th time you have 394L of dirty still in there.

 

[fatboyonabike]

as the fresh water is mixing with the dirty water everytime, you are diluting the dirty each time you add 10lts
I dont think its going to be a linear answer, and while you mince around changing 10lt at a time, the fish are still shitting in the water!

 

[moorey]

as the fresh water is mixing with the dirty water everytime, you are diluting the dirty each time you add 10lts
I dont think its going to be a linear answer, and while you mince around changing 10lt at a time, the fish are still shitting in the water!

Hence my homeopathy comment. You will always have the 'memory' of the original water, no matter what you do....

 

[No Skid Marks]

Hence my homeopathy comment. You will always have the 'memory' of the original water, no matter what you do....

I like dirty memories.
You'd have the same amount of dirty memories once all changed either way, and unprovable memory is incalculable as to its effect with either method, so back on track. Use the blue die theory if it sits better with your sensibilities Moorey.

 

[No Skid Marks]

and while you mince around changing 10lt at a time, the fish are still shitting in the water!

LOL, For that matter you could argue the filter is still filtering and then have to calculate hypothetically how much it's filtering Vs the amount of poo. But I never said there was still fish in there anyway. If there is or isn't a fish, it would be called Moorey. It may or may not be an eel.

 

[schred]

All in one go – take 200L out, replace with 200L of fresh water, so 400L of dirty water.

10L increments:

After the first one you have taken out 10/600 = 1.6667% of the dirty water, so there is 590L of dirty still in there
After the second one, taking another 1.66667% out, so now 580.16667L
…..
After the 25th time you have 394L of dirty still in there.

I get the same, and it does take into account the ongoing dilution. Hence why it takes 250 litres to change out the equivalent of 200L in one go.

 

[slippy]

But I never said there was still fish in there anyway.

In which case you would empty all the water out and replace, then you'd have 100% fresh clean water without all the dicking around.

 

[moorey]

In which case you would empty all the water out and replace, then you'd have 100% fresh clean water without all the dicking around.

Nope. Not meant to do that.

 

[fatboyonabike]

Nope. Not meant to do that.

you are if you want to fuck off all the fish and replace them with hermit crabs, but you will save 599.95 liters or water in doing so too

 

[No Skid Marks]

I get the same, and it does take into account the ongoing dilution. Hence why it takes 250 litres to change out the equivalent of 200L in one go.

but every new bucket changes the dilution ratio, as the dirty water is more diluted every new bucket.

 

[wesdadude]

For each swap you're removing 1.67% of the contents of the tank and then adding 10L of dye. Our base state is 600L fresh and 0L dye. Our First step is 590f+10d, 2nd: (590f + 10d)×0.983+10d).
The amount of dye is non linear. We can model the amount of dye using the recursive function

At n = 24 you have 199.2L of dye, and 205.8L at n = 25.

 

[moorey]

Use the blue die theory if it sits better with your sensibilities Moorey.

Sensibility? You're confusing me with someone else! :noidea:

 

[John U]

600 litres is a massive tank. Are you assuming completely uniform mixing after every bucket is added? If so surely that process would piss the fish off.

 

[cokeonspecialtwodollars]

Why not just have a fresh water supply trickle into the tank at one end and a drain at the high water level on the other, then the water is constantly replaced over a period of days/weeks/months as determined by the flow rate of the fresh water supply?

 

[Calvin27]

Mathematically, ever. It's an asypmtote. The tank concentration will never reach the fresh water concentration - it will keep approaching it but never reach. It's an asypmtote.

What you are asking is basically the same as asking: If a frog is 1m from the pond and jumps half the distance each jump, how many times will it take to get to the pond? Answer, never.

 

[s.dogg]

Never. It's an asypmtote. The tank concentration will never reach the fresh water concentration - it will keep approaching it but never reach. It's an asypmtote.

What you are asking is basically the same as asking: If a frog is 1m from the pond and jumps half the distance each jump, how many times will it take to get to the pond? Answer, never.

He didnt ask about getting it to compete fresh water.

 

[Calvin27]

He didnt ask about getting it to compete fresh water.

Misread, that. Working on it.