# A 30 cm piece of wire is cut in two. One piece is bent into the shape of a rectangle with a length-to-width of 2:1. What are the lengths the two pieces if the sum of the areas of the square and rectangle is a minimum?

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The other piece is bent into a square.

The other piece is bent into a square.

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Ananda Dasgupta
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Mar 29, 2018

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Let the width of the rectangle be

The total area now is

so that

To maximize the area, we must set

So, the length of the two pieces must be

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