Q. 29

Show that of all the rectangles with a given perimeter, the square has the largest area.

Answer :

Let us Consider a rectangle with length is x cm and breadth is y cm.


Since, Area of rectangle = length × breadth


A=xy …(i)


And, Perimeter of rectangle is , = 2(length + breadth)


P=2(x + y) …(ii)


Now, From (ii),



Putting the value of y in equation (ii), we get




Now, differentiate A, w.r.t x , we get




If then




If


Since,


Corresponds to a maximum value of A.


The area is maximum when the sides are


Hence, This is the square, with largest area.


Hence, Proved


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