Format Neutral Challenge

[SEANT]

Wow.

 

I sure do like to complicate things.

. . . .

 

With that post it looks like you have been trying hard to simplify things.

 

I definitely considered the notion of making multiple links, but wasn’t sure how conspicuous I wanted to be if the challenge “fell on its face”. Limited number of participants notwithstanding, the quality of the results guarantee this thread's success.

 

These type of challenges may be doomed to only a few participants due to the fact that the first viable solution virtually eliminates all mystery. New arrivals will undoubtedly read through the thread to see if the effort is still worthwhile, and/or they would not bother adding a response if their method is too similar to an answer already posted.

 

It may have helped to have used a similar setup as the Competitions in the Community forum. There the Challenge is described in one thread - Each participants adds a link in that thread to there submission in another. At least then a new participant could avoid seeing a solution that would quell their own enthusiasm.

[eldon]

Very nice solution Lee Mac.

 

I suppose that the shape of the area being a cyclic quadrilateral is a red herring, and that Ptolemy's Theorem does not lead to any simplification.

 

Oh well, back to the drawing board

[SEANT]

Very nice solution Lee Mac.

 

I suppose that the shape of the area being a cyclic quadrilateral is a red herring, and that Ptolemy's Theorem does not lead to any simplification.

 

Oh well, back to the drawing board

 

 

Damn! If my intention for this thread was to inspire some thought, then I think it was successful. I know I have some more reading to do.

[Lee Mac]

With that post it looks like you have been trying hard to simplify things.

 

After finding that simple expression for the area of the oversized triangle, the problem just seemed to be crying out for a concise and elegant solution.

 

It may have helped to have used a similar setup as the Competitions in the Community forum. There the Challenge is described in one thread - Each participants adds a link in that thread to there submission in another. At least then a new participant could avoid seeing a solution that would quell their own enthusiasm.

 

I think that's a good way to do it, reminiscent of the 'Ferris Euler' threads in the C-Programming board over at theSwamp.

 

Very nice solution Lee Mac.

 

I suppose that the shape of the area being a cyclic quadrilateral is a red herring, and that Ptolemy's Theorem does not lead to any simplification.

 

Thanks Eldon

 

I'd hoped there was a 'trick' to be seen with the area so that the ugly trig method needn't be used, but as yet I haven't found it...

[Lee Mac]

This is the result of plotting the Area as a function of radius, using the function from my second solution.

 

Area-Radius Plot

 

[/url]This demonstrates the values at which the maximum area is reached:

 

Local Maxima

 

Within the range 0

 

r = 1/21*(7-2*sqrt(7)) 

[eldon]

This is the result of plotting the Area as a function of radius, using the function from my second solution.

 

r = 1/21*(7-2*sqrt(7)) 

 

Thank you for taking the time and effort to post all that, and the beautifully simple equation for the radius at the maximum area.

[SEANT]

That is beautifully simple, though yikes, those are some scary calculations getting there. I guess I’m fortunate that my Calculus is so rusty I didn’t even start an attempt on my own.*

 

Earlier in this thread I mentioned that I’d post my effort, and here it is. This was an initial attempt, prior to noticing any of the other “Outcropped Triangle” relationships. Nothing new to add other than that it is in a different format (Excel spreadsheet).

 

If I have any math intensive programming, I tend to prototype the calculations in Excel. It is particularly useful for discovering non-intuitive relationships. Is that a common practice? How many of you other programmers keep an session of Excel running in the background?

 

*Though I did, at least, dig out my old Calculus text book.

AreaCalc.zip

Edited June 8, 2011 by SEANT
Added comment

[SEANT]

Incidentally, here is the post that inspired the challenge (the link may require membership to view).

 

I played around by adding some additional parameters – I’m glad I did.

http://www.physicsforums.com/showthread.php?t=502058

[Lee Mac]

That is beautifully simple, though yikes, those are some scary calculations getting there. I guess I’m fortunate that my Calculus is so rusty I didn’t even start an attempt on my own.*

 

It is indeed a beautifully simple result, but as you say, the calculation of the derivative of the expression is far from simple - one of the reasons I was striving to avoid using the lengths of the oversized triangle and hence avoid the nasty square rooots...

 

Earlier in this thread I mentioned that I’d post my effort, and here it is. This was an initial attempt, prior to noticing any of the other “Outcropped Triangle” relationships. Nothing new to add other than that it is in a different format (Excel spreadsheet).

 

I really like how you've formatted the calculations, with everything fully labelled and described, the thought process is much easier to follow than the explanations in my posts.

 

If I have any math intensive programming, I tend to prototype the calculations in Excel. It is particularly useful for discovering non-intuitive relationships. Is that a common practice? How many of you other programmers keep an session of Excel running in the background?

 

I tend to derive everything on pen and paper first, since its much easier to express and manipulate the algebra and see what's going on - also I find you are more likely to see patterns and places where the expression can be simplified.

[SEANT]

. . . .

 

I tend to derive everything on pen and paper first, since its much easier to express and manipulate the algebra and see what's going on - also I find you are more likely to see patterns and places where the expression can be simplified.

 

 

That’s true. A fair bit of formatting/algebra has to be is written out first to know what to plug in to Excel.

 

 

 

Workflow and methodology probably vary a lot from one programmer to another. Many of my early tasks, when I first sat in front of a computer, involved Excel. I may have “imprinted” on the app, thus developing an unnatural fondness.

[Lee Mac]

Workflow and methodology probably vary a lot from one programmer to another. Many of my early tasks, when I first sat in front of a computer, involved Excel. I may have “imprinted” on the app, thus developing an unnatural fondness.

 

It is definitely my favourite of the Office applications