Q. 93.8( 4 Votes )
Given the sum of the perimeters of a square and a circle, show that the sum of their areas is least when one side of the square is equal to diameter of the circle.
Answer :
Let us say the sum of perimeter of square and circumference of circle be L
Given: Sum of the perimeters of a square and a circle.
Assuming, side of square = a and radius of circle = r
Then, L = 4a + 2πr …1
Let the sum of area of square and circle be S
So, S = a2 + πr2
S =
Condition for maxima and minima
= 0
So, for
>0
This is the condition for minima
From equation 1
Substituting from equation 2
a = 2r
Hence, proved.
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