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January 26th, 2010, 12:58 AM  #1 
Member Joined: May 2009 Posts: 90 Thanks: 0  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]26012010 094403_0043.jpg[/attachment:24fin3ph] 
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. = 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: 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 is a (modified) discriminant, which tells you what your next step should be. For quadratics, it's b²4ac, 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: 20,636 Thanks: 2080 
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: 20,636 Thanks: 2080 
Rearranging the cubic equation gives h = (305.73  h³)/87.48. Evaluating the righthand 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 
Member Joined: May 2009 Posts: 90 Thanks: 0  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] 

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