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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]
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January 26th, 2010, 04:54 AM   #2
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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: h + 87.48h - 305.73 = 0

Step 1. Use a transformation to put it into the reduced form x+3px+2q = 0 with no x term.

Already there, so we don't have to do anything. p = 29.16 and q = -152.865

Step 2. Find the discriminant p+q.

= p+q
= 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.
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January 26th, 2010, 07:46 AM   #3
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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.....
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January 26th, 2010, 08:40 AM   #4
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Re: Need help with analytical method

h + 87.48h - 305.73 = 0

"Step 1. Use a transformation to put it into the reduced form x+3px+2q = 0 with no x 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 b-4ac, but for cubics it's p+q (or something similar).
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January 26th, 2010, 09:13 AM   #5
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Re: Need help with analytical method

I get it

Thanks...Mark
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January 27th, 2010, 12:03 AM   #6
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Re: Need help with analytical method

I understand that now but how exactly do I key the last bit into a calculator?
Thanks....
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January 27th, 2010, 03:29 AM   #7
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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).
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January 27th, 2010, 04:07 AM   #8
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Re: Need help with analytical method

Hi Skipjack,
which other ways are you referring to? Iteration has been suggested; how does this work?

Thanks....
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January 27th, 2010, 06:19 AM   #9
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Rearranging the cubic equation gives h = (305.73 - h)/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.
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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]
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File Type: jpg table.jpg (240.0 KB, 136 views)
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