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January 26th, 2010, 12:58 AM   #1
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Need help with analytical method

Hi,

In the attachment I have arrived at the correct answer but my tutor tells me there are mistakes in the last part of my analytical method as shown in the box.

Can someone have a look and tell me whats wrong with it.

Thanks...Mark[attachment=0:24fin3ph]26-01-2010 09-44-03_0043.jpg[/attachment:24fin3ph]
Attached Images
 26-01-2010 09-44-03_0043.jpg (135.3 KB, 169 views)

 January 26th, 2010, 04:54 AM #2 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: Need help with analytical method The problem is that you have plucked the solution out of thin air. None of the numbers 30.98, 274.8 or 3.14 are supported by any working. You start with this cubic equation: h³ + 87.48h - 305.73 = 0 Step 1. Use a transformation to put it into the reduced form x³+3px+2q = 0 with no x² term. Already there, so we don't have to do anything. p = 29.16 and q = -152.865 Step 2. Find the discriminant p³+q². $\Delta$ = p³+q² = 24794.911296 + 23367.708225 = 48162.619521 > 0 The discriminant is positive, so there is exactly one real root and we can use Cardano's formulae. Step 3. Apply Cardano's formulae. You're only interested in the real root: $h= \sqrt[3]{\sqrt{\Delta}-q} + \sqrt[3]{-\sqrt{\Delta}-q}$ Step 4. Reverse the original transformation. No need in this case.
 January 26th, 2010, 07:46 AM #3 Member   Joined: May 2009 Posts: 90 Thanks: 0 Re: Need help with analytical method Hi, Thanks for the reply Can you be a bit more specific,this is my first attempt. Why the -152.865 And explain delta Thanks.....
 January 26th, 2010, 08:40 AM #4 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: Need help with analytical method h³ + 87.48h - 305.73 = 0 "Step 1. Use a transformation to put it into the reduced form x³+3px+2q = 0 with no x² term." Equating coefficients: 3p = 87.48 and 2q = -305.73 Hence p = 29.16 and q = -152.865 $\Delta$ is a (modified) discriminant, which tells you what your next step should be. For quadratics, it's b²-4ac, but for cubics it's p³+q² (or something similar).
 January 26th, 2010, 09:13 AM #5 Member   Joined: May 2009 Posts: 90 Thanks: 0 Re: Need help with analytical method I get it Thanks...Mark
 January 27th, 2010, 12:03 AM #6 Member   Joined: May 2009 Posts: 90 Thanks: 0 Re: Need help with analytical method I understand that now but how exactly do I key the last bit into a calculator? Thanks....
 January 27th, 2010, 03:29 AM #7 Global Moderator   Joined: Dec 2006 Posts: 19,190 Thanks: 1649 Were you expected to solve the cubic equation "analytically"? Other methods, based on estimating the solution, are also available (and easy to do if a highly precise value is not required).
 January 27th, 2010, 04:07 AM #8 Member   Joined: May 2009 Posts: 90 Thanks: 0 Re: Need help with analytical method Hi Skipjack, which other ways are you referring to? Iteration has been suggested; how does this work? Thanks....
 January 27th, 2010, 06:19 AM #9 Global Moderator   Joined: Dec 2006 Posts: 19,190 Thanks: 1649 Rearranging the cubic equation gives h = (305.73 - h³)/87.48. Evaluating the right-hand side of that equation for some approximate value of h gives a new approximation. Starting with 3, successive results are (to 3 decimal places) 3.186, 3.125, 3.146, 3.139, 3.141, etc.
January 27th, 2010, 07:30 AM   #10
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Re: Need help with analytical method

Hi,

I got (305.73 - h3)/87.48

Est 3 = (305.73 - 27)/87.48 = 3.186213992

After the initial 3.18, what keys do I hit next, or do you work it out on paper/grid etc?
What's the next step to progress to 3.14 and how do I know when I have arrived at the final answer with this method? I have attached a table; is that what you mean?

Thanks....Mark[attachment=0:3alvfnxu]table.jpg[/attachment:3alvfnxu]
Attached Images
 table.jpg (240.0 KB, 136 views)

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