Q. 245.0( 3 Votes )

Show that of all the rectangles of given area, the square has the smallest perimeter.

Answer :

Let us Consider a rectangle with length is xcm 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 (i),



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



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




If then



x2=A



If


Since,


Therefore,



Therefore, A rectangle with given area will have atleast perimeter when x=y or it is a square.


Hence, Proved


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