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AdrianCavinder · Jan 31, 2012, 07:49 AM

HI again. Sorry, I have another problem I canÃ¢â‚¬â„¢t seem to work out:
A funnel has a circular top of diameter 20cm and a height of 30cm. When the depth of liquid in the funnel is 12cm, the liquid is dripping from the funnel at a rate of 0.2cm^3s^-1. At what rate is the depth of the liquid in the funnel decreasing at this instant?
Now, from the chain rule, I understand that dV/dt = dV/dh x dh/dt, where V = Volume h = depth and t = time.
From the formula, V = (1/3)Ãâ‚¬r^2h, dV/dh = (1/3)Ãâ‚¬r^2, so 0.2 = (1/3)Ãâ‚¬r^2 x dh/dt. As the depth is twelve, I believe the radius to be 8 because of the proportion rule 30:12 = 20:8
So dh/dt = 0.6/64Ãâ‚¬ = 0.00298cms^-1, but the answer is 0.004cms^-1.
Can you show me where IÃ¢â‚¬â„¢m zagging when I should be zigging? Cheers.

ebaines · Jan 31, 2012, 08:07 AM

First - please note that you cannot copy and paste math formulas into this web site, as special characters don't appear correctly.

You have
$ V = \frac 1 3 \pi r^2 h $

where r and h are related by:

$ r = \frac 2 3 h $

So:

$ V = \frac 1 3 \pi (\frac 2 3 h)^2 h = \frac 4 {27} \pi h^3 $

Now you can use the change rule:

$ \frac {dV}{dt} = \frac {dV} {dh} \frac {dh} {dt},\\ \frac {dh}{dt} = \frac {\frac {dV}{dt}} {\frac {dV}{dh}} $

Can you take it from here?

AdrianCavinder · Jan 31, 2012, 08:52 PM

Thanks, that was the 'zagging' bit. By substituting r = (1/3)h I get the correct answer.
Btw, I don't copy-paste math formulas, but I have noticed that pi(Ï€) looks like pi on my laptop but all sorts of funny stuff when posted. Even a simple apostrophe ' looks weird. Is it because I am using a mac? Or can you let me know what to do?
Anyway, thanks so much for your help: much appreciated.

ebaines · Feb 1, 2012, 06:47 AM

Originally Posted by AdrianCavinder

By substituting r = (1/3)h I get the correct answer.

Doh! Yes, I should have written r = (1/3)h, not r= (2/3)h. Glad you caught that.

Originally Posted by AdrianCavinder

Btw, I don't copy-paste math formulas, but I have noticed that pi(Ï€) looks like pi on my laptop but all sorts of funny stuff when posted. Even a simple apostrophe ' looks weird. Is it because I am using a mac? Or can you let me know what to do?

I've used both mac and PC without issue - just stick to the "regular" keyboard characters. What did you do to try to get the pi symbol?

AdrianCavinder · Feb 1, 2012, 07:49 AM

Hi.
Well, I used option P for pi. I also use an American keyboard if that makes a difference. If you have a solution, I would appreciate the input. Thanks.

ebaines · Feb 1, 2012, 08:05 AM

Just use the regular keyboard - no characters that require option, control, alt, etc.

If you're interested in learning how to embed math symbols, check out the LaTeX tutorial here:
https://www.askmehelpdesk.com/mathem...as-612918.html