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September 11th, 2013, 05:49 PM   #1
Doc
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Optimisation problem

I am really stuck on this problem:

A straight piece of wire 12cm in length is to be cut into two pieces. One piece is bent to form a circle and the other a square. Where should the cut be made so that the combined area of both shapes:

a.minimum
b. maximum


Thank you for your help
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September 11th, 2013, 07:00 PM   #2
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Re: Optimisation problem

I did a few calculations and found the minimum will occur if you cut at the and use that length for the circle.

The maximum is a trick question , i think. Use all 12 cm to make a circle. Any piece you cut off to use in making a square will diminish the sum total of the area.

Hope someone can confirm or deny my result since i'm very tired right now to write it up in LaTeX.

Essentially i set it up 2 ways





A1 uses x for the square , A2 uses x for the circle. Took some derivatives and found A2 gave exactly the same area as A1 sum total so doing it twice was un-necessary , maybe you can check this (if i didn't make any errors , i didn't confirm the minimum myself)

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September 11th, 2013, 07:51 PM   #3
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Re: Optimisation problem

You are correct. See this post here:

http://mathhelpboards.com/questions-...html#post23754

For my derivations (using several methods) of the formulas (where is the length of the wire) to minimize the total area:

Amount to be used for square:



Amount to be used for circle:



You are also correct that the maximum comes from using all of the wire for the circle.
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September 11th, 2013, 08:11 PM   #4
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Re: Optimisation problem

Thanx [color=#BF0000]MarkFL[/color] , very helpful link , professionally done. I'm glad i didn't log off when i noticed you were browsing the forum ... will sleep easier knowing i got the right answer ... agentredlum ... signing off.

P.S. One of these days i'm going to learn how to use the Lagrange methods you incorporate often , they seem very helpful.

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September 12th, 2013, 02:02 AM   #5
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Re: Optimisation problem

Thanks very much both of you for helping me. I always get stuck on question like this, does anyone know of a way for me to overcome this problem?
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