Boas ch1 ex12--the water purification question

[vecsen]

Homework Statement

The question is : "In a water purification process, one-nth of the impurity is removed in the first stage.
In each succeeding stage, the amount of impurity removed is one-nth of that removed in the preceding stage. Show that if n = 2, the water can be made as pure as you like, but that if n = 3, [I take this to mean that 66.667% of impurities remaining--] at least one-half of the impurity will remain no matter how many stages are used. "

Homework Equations

I am thinking, granted, there will always be some residue, but that is what 'clean water is': one with only traces of other substances.

I tried at first on paper (with squares). The residue becomes vanishingly small.

As I mention in the title, I expect to be shown in error, but this seems monstrously improbable to me. For instance how is this different from, say 2% inflation? Suppose that--to use gov. statistics-- 98% on average of the value of money remains from year to year. After a century money will have lost ~95.245% of its initial value (there is a certain degree of realism here :).

The Attempt at a Solution

In short I am thinking more along the lines of (1-1/n)^k rather than a *( (1-r^k)/ 1-r). If by any chance I am right, whatever was that woman thinking of?

Any assistance would be greatly appreciated

 

[gneill]

Say you start with 1 unit of impurity. The first stage removes 1/n of that so you're left with (1 - 1/n). The next stage removes 1/n of the amount of impurity removed in the preceding stage. What amount of impurity was removed in the preceding stage? What's 1 n^th of that?

 

[vecsen]

thanks for the time & answer.
However, you mention "the next stage removes 1/n of the amount of impurity removed in the preceding stage" . I beg to differ: to my mind stage k removes 1/n of the amount REMAINING dissolved in the medium. The way I think of it, you are pointing to (1/n^k), while I am saying [(n-1)/n] ^k.
I hope I am not making a nuisance of myself & thanks for your time & patience
regards,
v

 

[gneill]

The problem specifically says,

"In each succeeding stage, the amount of impurity removed is one-nth of that removed in the preceding stage."

What was removed was 1/n.

 

[vecsen]

Thank you, and thanks for your patience. Now I understand the question & the suggestion that unless 50% is removed... Grateful.
Regards,
v.

 

[gneill]

You're quite welcome.

Cheers.

 

[SammyS]

The problem specifically says,

"In each succeeding stage, the amount of impurity removed is one-nth of that removed in the preceding stage."

What was removed was 1/n.

Thanks g. !

I'm glad that you read this so carefully. I nearly responded to this earlier and totally missed this detail !

 

[gneill]

Thanks g. !

I'm glad that you read this so carefully. I nearly responded to this earlier and totally missed this detail !

No problem. I'm lucky I didn't miss it, too!