September 11th, 2013, 05:49 PM  #1 
Member Joined: Aug 2012 Posts: 71 Thanks: 1  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 
September 11th, 2013, 07:00 PM  #2 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  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 unnecessary , maybe you can check this (if i didn't make any errors , i didn't confirm the minimum myself) 
September 11th, 2013, 07:51 PM  #3 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  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. 
September 11th, 2013, 08:11 PM  #4 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234  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. 
September 12th, 2013, 02:02 AM  #5 
Member Joined: Aug 2012 Posts: 71 Thanks: 1  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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