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