Let S be the set of all rectangles with perimeter P. Showthat the square is the element of S with largest area.I know that A = x ( ( P 2 ) − x ) , but the next steps is ( P 4 ) 2 − ( x − ( P 4 ) ) 2 , which I dont get.

Question

Let S be the set of all rectangles with perimeter P. Showthat the square is the element of S with largest area.I know that   A  =  x  (  (      P    2    )  −  x  )  ,   but the next steps is   (      P    4        )    2    −  (  x  −  (      P    4    )      )    2  , which I dont get.

illpnthr21vw · Accepted Answer

I am assuming that you have a set of solutions in front of you so iwon&#039;t give the whole solution (please say if you don&#039;t and i willgive the rest of the solution)So you are at the point   A  =  x  (  (      P    2    )  −  x  )When you expand this out you get  A  =  −      x    2    +  (      P    2    )  xTake out a factor of -1  A  =  −  1  (      x    2    −  (      P    2    )  x  )Now we have to do something called completing the square.You take the co-efficient of x, divide it by two then squareit.Finally you add it and take it away from the equation(You will see what i mean)  A  =  −  1  (      x    2    −  (      P    2    )  x  +  (  (      P    2    )      /    2      )    2    −  (  (      P    2    )      /    2      )    2    )  &amp;lt;  b  r  &amp;gt;  A  =  −  1  (      x    2    −  (      P    2    )  x  +  (      P    4        )    2    −  (      P    4        )    2    )Now if we look at the first three terms on the right,       x    2    −  (      P    2    )  x  +  (      P    4        )    2  They are a perfect square and we can factorise them as       x    2    −  (      P    2    )  x  +  (      P    4        )    2    =  (  x  −  (      P    4    )      )    2  Now we sub that back into the equation  A  =  −  1  (      x    2    −  (      P    2    )  x  +  (      P    4        )    2    −  (      P    4        )    2    )    A  =  −  1  (  (  x  −  (      P    4    )      )    2    −  (      P    4        )    2    )    Now expand out the -1 and hey presto you get    A  =  (      P    4        )    2    −  (  x  −  (      P    4    )      )    2

Let S be the set of all rectangles with perimeter P. Showthat the square is the element of S with largest area. I know that A = x((P\2)-x), but the next steps is (P/4)^2 -(x-(P/4))^2, which I dont get.

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