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.

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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