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A wire of length 36 m is cut into two pieces one of the pieces is bent the form a square and the other is bent to form a circle of the sum of the areas of the figures is minimum and the circumference of the circle is k (meter) then $(\frac{4}{π} + 1) k$ is equal to .......

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answer is 36.

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Detailed Solution

$x + y = 36$ Let x be the perimeter of a square and y be the perimeter of a circle
Side of a square $= \frac{x}{y}$ radius $= \frac{y}{2 π}$ Sum of Areas

$= {(\frac{x}{y})}^{2} + π {(\frac{y}{2 π})}^{2} = \frac{x^{2}}{16} + \frac{{(36 - x)}^{2}}{4 π}$

Formin Area $x = \frac{144}{π + 4} \Rightarrow$

radius $y = 36 - \frac{144}{π + 4}$

$K = \frac{36 π}{π + 4}$

$(\frac{4}{π} + 1) K = 36$

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