A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then ( 4 / pi + 1 )k is equal to _____________.

Let x + y = 36x is perimeter of square and y is perimeter of circle side of square = x/4radius of circle = y / 2pi Sum Areas = ( x / 4 )^2 + pi ( y / 2pi )^2 = x^2 / 16 + (36 - x)^2 / 4pi For min Area :x = 144 / pi + 4 => Radius = y = 36 - 144 / pi + 4 => k = 36pi / pi + 4 ( 4 / pi + 1 )k = 36

A wire of length 36 m is cut into two pieces one of the... | null (Mathematics)

Let

x + y = 36

x

is perimeter of square and y is perimeter of circle side of square

= x/4

radius of circle =

{y \over {2\pi }}

Sum Areas =

{\left( {{x \over 4}} \right)^2} + \pi {\left( {{y \over {2\pi }}} \right)^2}

= {{{x^2}} \over {16}} + {{{{(36 - x)}^2}} \over {4\pi }}

For min Area :

x = {{144} \over {\pi + 4}}

\Rightarrow \text{Radius} = y = 36 - {{144} \over {\pi + 4}}

\Rightarrow k = {{36\pi } \over {\pi + 4}}

\left( {{4 \over \pi } + 1} \right)k = 36

Explanation

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