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Maximize and Minimize a 12" piece of wire into a square and circle

A piece of string length L is to be cut into two pieces. One piece is formed into a square, the other into a circle. Where should the string be cut in order for the total are eclipsed by the two shapes be as small as possible. And how can they be cut to make the total area enclosed as big as possible.

This is for Calculus, so I am guessing it will be a minimising and maximising question, but I can't remember how to do it. And advice?

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marked as duplicate by Rahul, Chris Eagle, t.b., Gerry Myerson, Zev Chonoles Jun 6 '12 at 19:54

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question. this contains similar question solved – Bhargav Jun 4 '12 at 14:45
Please see this MSE question. And it is probably not the only one! – André Nicolas Jun 4 '12 at 14:47

wellcome to math stackexchange.

Let $x$ length of the cut. if we build the square with the first part $x$ and the circle with the second part $L-x$, the area of the square would be: $(\frac{x}{4})^2$ and Furthermore, as the length of the circle would be $L-x=2\pi r$ then $r =\frac{L-x}{2\pi}$. The function must be optimized is:

$$f(x)=\left(\frac{x}{4}\right)^2+ \pi \left(\frac{L-x}{2\pi}\right)^2$$

Can you find a minimum? the maximum would be full length to build the square or circle.

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