My Math Forum Optimisation problem

 Calculus Calculus Math Forum

 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 $\ x \= \ \frac{12 \pi}{ \pi \ + \ 4} \ \approx \ 5.27881 \ cm \ \$ 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 $A_1 \= \ \frac{x^2}{16} \ + \ \frac{(12 \ - \ x)^2}{4 \pi}$ $A_2 \= \ \frac{x^2}{4 \pi } \ + \ \frac{(12 \ - \ x)^2}{16}$ 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)
 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 $L$ is the length of the wire) to minimize the total area: Amount to be used for square: $S=\frac{4L}{4+\pi}$ Amount to be used for circle: $C=\frac{\pi L}{4+\pi}$ 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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### a wire of length 25th is to be cut into two pieces one of th pieces is to be made into a square and other into a circle

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