Question

Please help me solve this!
A tank is filled with 100L of a 40% alcohol solution (by volume) you repeatedly perform the following: remove 2 L of the soln from the tank and add 2L of 10 % alcohol soln
A. let Cn be a concentration of the soln in the tank after the nth replacement where C0 =40%
write the first five terms of {Cn}
B. after how many replacements does the alcohol concentration reach 15%?
thank you in advance!

Answer 1

Ok, well it seems like we have a recursive relation here. What have you been able to do with this problem so far?

Answer 2

well I came up with Cn=40 - o.60n

Answer 3

I don't think its correct

Answer 4

Sure, can you write out how you found that solution?

Answer 5

ok

Answer 6

c1=40L- 2L*40%+2L*10%= a1
c2= a1-2L*40%+2L*10%
c3= a2-*40%+2L*10%
cn=40L-n(2(40%-10%))

Answer 7

Ok, so first off the tank is 100L, not 40L
and to simplify you could write 100L @40% - 2L @40% as 98L @ 40%
then you add 2L at 10% correctly, but what is your new concentration going to be?

Answer 8

but 100 L is 40% alcohol so i multpiplied

Answer 9

What you have there is *total* ethanol (in L), not concentration.
so you want to divide by the total volume, yes?

Answer 10

yes

Answer 11

Right, so 100L x 0.4 = 40L
Ah, oh, hold on

Answer 12

Oh, it looks like you are figuring out total volume of ethanol...
You can do it that way too. :D

Answer 13

but i dont understand

Answer 14

Well, I suggest this:
because it asks when the concentration drops below 15%, we should write a recursive relation for the concentration.
just so that for each step we don't need to do an additional calculation to find the concentration.
how does that sound?

Answer 15

yah ok its a little challenging for me sorry

Answer 16

No worries. What I meant was, the expression you wrote would be correct if you were looking at just the total volume of ethanol. (how many of the 100L is ethanol).
The way I suggested we do it is that we don't look at the *amount* but the *percentage of total* that is ethanol... because that is what the question is asking about. :)

Answer 17

ok

Answer 18

Ok, so if we are to rewrite our recursive relationship in terms of %ethanol, we might end up with something like....