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So you're moving into your new apartment, and you're trying to bring your sofa. The problem is, the hallway turns and you have to fit your sofa around a corner. If it's a small sofa, that might not be a problem, but a really big sofa is sure to get stuck. If you're a mathematician, you ask yourself: What's the largest sofa you could possibly fit around the corner? It doesn't have to be a rectangular sofa either, it can be any shape.

 Aug 29, 2017
 #1
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The largest shape I could conceive is a semicircle. 

 

If the corridor is x units long, then the semicircle has the area of \(\frac{\pi x^2}{2}\). This is better than a square.

 Aug 29, 2017

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