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Alike but different dimensions

Hello fellow humans. My confusion is on the dimensions of objects. I noticed that if 3 shapes are made, A, B and C, all quadrilaterals. A is a 5x5, B is a 6x4 and C is a 8x2. The perimeter of all shapes are the same, 20, but their areas differ. Please indulge my inquisitiveness.

Note by Ibukunoluwa Abiodun
4 years, 5 months ago

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Well it means that 2(l+b)=20 l+b=10. But this does not mean, that the area , lb, is fixed. l=8, b=2, gives lb=16. l=5,b=5 gives lb=25. l=6,b=4 gives lb=24. Shourya Pandey · 4 years, 5 months ago

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@Shourya Pandey But why are their areas different in such a way that the area tends to be more towards the square shapes :) Ibukunoluwa Abiodun · 4 years, 5 months ago

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@Ibukunoluwa Abiodun That is because: Suppose l>=b, without loss of generality. Say l=5+d. then b=5-d So lb=(5+d)(5-d)=25-d^2. So as l and b come closer to resemble a square, i.e., l=b=5, then d becomes smaller and smaller and the area increases. Shourya Pandey · 4 years, 5 months ago

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@Shourya Pandey Another way is the A.M.-G.M. inequality, stating \(\frac {l+b}{2}\)>= \(\sqrt lb\) So that lb<=25, with equalty iff l=b=5, i.e., the figure is a square. Shourya Pandey · 4 years, 5 months ago

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@Shourya Pandey Thanks a lot, now I have full light to my darkness, thanks to you. Man you are good :) Ibukunoluwa Abiodun · 4 years, 5 months ago

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