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 Maths Problem
 B737200 From Malta, joined Feb 2005, 225 posts, RR: 2Posted Tue Jan 19 2010 04:38:12 UTC (5 years 2 months 1 week 5 days 12 hours ago) and read 2039 times:

 Hi, I have a maths problem and since I've seen people asking all sorts of questions over here I thought "why not?" So I need to solve this equation: 2.64 = (a) + ln(a) where a is the variable i'm looking for and ln is the function ln [log_e]. The only thing I could come up with is to insert values of a and repeat the process correcting a as I go. Anything better than iteration? Thanks.
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 11 replies: All unread, jump to last
 Oly720man From United Kingdom, joined May 2004, 6945 posts, RR: 11 Reply 1, posted Tue Jan 19 2010 05:01:58 UTC (5 years 2 months 1 week 5 days 12 hours ago) and read 2024 times:

 Is this in any particular bit of maths? One solution would be to use a series solution for ln(a) which would give a quadratic/cubic you could solve. http://en.wikipedia.org/wiki/Natural...arithm#Derivative.2C_Taylor_series
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 B737200 From Malta, joined Feb 2005, 225 posts, RR: 2 Reply 2, posted Tue Jan 19 2010 06:13:06 UTC (5 years 2 months 1 week 5 days 11 hours ago) and read 1982 times:

 Its actually a fluids problem but I got stuck on the mathematical part. The problem has water flowing out of a tank through an orifice whilst water is also flowing into the tank. I ended up with an integral which I solved using a given result and I ended up with the above equation. It should be correct since my calculator worked it out and the answer matched but our lecturer is a bit fussy so I'm not sure if he'd except that in an exam. I'll take a look at the link you sent, it seems somewhat familiar, maybe we did it last year in our maths unit. Hopefully something like that would be in the equation sheet we get, good luck trying to remember all the funny mathematical solutions to stuff that MIGHT come out. I get the feeling he is going to test us on mathematics as much us he is going to test us on our fluid's knowledge. What nudged me towards the iteration method was the fact that the question states the we are to give an approximate answer; of course using a series solution would also yield and approximate value so maybe this is it. Thanks, series totally slipped my mind, maybe its because I can't stand them
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 BMI727 From United States of America, joined Feb 2009, 16065 posts, RR: 27 Reply 3, posted Tue Jan 19 2010 09:04:21 UTC (5 years 2 months 1 week 5 days 8 hours ago) and read 1906 times:

 Quoting B737200 (Reply 2):The problem has water flowing out of a tank through an orifice whilst water is also flowing into the tank

Perhaps looking at the differential equations for a mixing problem would be of help.

 Why do Aerospace Engineering students have to turn things in on time?
 Yellowstone From United States of America, joined Aug 2006, 3071 posts, RR: 4 Reply 4, posted Tue Jan 19 2010 14:31:58 UTC (5 years 2 months 1 week 5 days 2 hours ago) and read 1805 times:

 Quoting B737200 (Reply 2):What nudged me towards the iteration method was the fact that the question states the we are to give an approximate answer; of course using a series solution would also yield and approximate value so maybe this is it.

What you want to use here is something called Newton's method.

http://en.wikipedia.org/wiki/Newton%27s_method

First, rewrite your equation so that it's equal to zero. f(a) = a + ln(a) - 2.64 = 0

Next, you pick some value of a, call it a_0, that is decently close to the right answer. Then, calculate the following, where f ' (a_0) is the derivative of the function f evaluated at a_0.

a_1 = a_0 - ( f(a_0) / f ' (a_0) )

Calculate a_2 in the same fashion, replacing a_0 with a_1. Repeat until a_n converges.

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 Fabo From Slovakia, joined Aug 2005, 1219 posts, RR: 1 Reply 5, posted Tue Jan 19 2010 16:42:44 UTC (5 years 2 months 1 week 5 days ago) and read 1752 times:

 Why repeat? If I understood right, you can use limit. You should get to about 1.96467 if I am not mistaken. I will try figuring out how to isolate a, but do not expect much from me right now...
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 B737200 From Malta, joined Feb 2005, 225 posts, RR: 2 Reply 6, posted Tue Jan 19 2010 23:53:44 UTC (5 years 2 months 1 week 4 days 17 hours ago) and read 1690 times:

Firstly thanks for all the input, I wasn't expected this much help for my little mathematical conundrum.

 Quoting Yellowstone (Reply 4):What you want to use here is something called Newton's method.

I was actually studying it this week, when I was trying the fluids problem I hadn't really gone over my maths note yet. Seems to bring a bit of order to my "lets just throw in some values" method.

 Quoting Fabo (Reply 5):Why repeat? If I understood right, you can use limit. You should get to about 1.96467 if I am not mistaken

By using a limit you mean with Newton's method? The version we learned didn't use that, I'll look around but if I misunderstood you I'll be on a wild goose chase for nothing.

 Quoting Fabo (Reply 5):I will try figuring out how to isolate a, but do not expect much from me right now...

Don't worry if you're busy, I understand and thanks for your input.

 Quoting BMI727 (Reply 3):Perhaps looking at the differential equations for a mixing problem would be of help.

I derived a differential equation. The one I derived was:

Q_out - Q_in = A_tank * (dh/dt) where Qs are volume flow rates, A is area and h is the height of water level in the tank.

On working it out using a given result printed under the question itself I ended up with the 2.64 = lna +a.

Not sure if you are referring to the same thing.

Once again thanks a million, very helpful.

 Lady Guinness is ready to fly...
 Ag92 From India, joined Jul 2006, 1317 posts, RR: 0 Reply 7, posted Wed Jan 20 2010 03:50:12 UTC (5 years 2 months 1 week 4 days 13 hours ago) and read 1663 times:

 I don't know if I will be of any help, but I would just graph it, and calculate the value through a graph
 Flighty From United States of America, joined Apr 2007, 9001 posts, RR: 3 Reply 8, posted Wed Jan 20 2010 04:59:20 UTC (5 years 2 months 1 week 4 days 12 hours ago) and read 1650 times:

 Use my very most favoritest website ever: http://www.wolframalpha.com/input/?i=2.64%3D+a+%2B+ln%28a%29 You're welcome.
 Fabo From Slovakia, joined Aug 2005, 1219 posts, RR: 1 Reply 9, posted Wed Jan 20 2010 06:20:50 UTC (5 years 2 months 1 week 4 days 10 hours ago) and read 1625 times:

 Flighty: That is what I used to come to 1.96467 I mentioned   Nice web, for sure.
 The light at the end of tunnel turn out to be a lighted sing saying NO EXIT
 Waterpolodan From United States of America, joined Feb 2005, 1649 posts, RR: 4 Reply 10, posted Wed Jan 20 2010 08:16:04 UTC (5 years 2 months 1 week 4 days 9 hours ago) and read 1601 times:

 Having Maths trouble? Might I suggest this video as a useful tool- http://www.youtube.com/watch?v=Pj2NOTanzWI
 B737200 From Malta, joined Feb 2005, 225 posts, RR: 2 Reply 11, posted Wed Jan 20 2010 08:54:44 UTC (5 years 2 months 1 week 4 days 8 hours ago) and read 1590 times:

 Thanks for the website, its now in my favorites.
 Lady Guinness is ready to fly...
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