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# Tell me? A wire of length 2 units is cut into two parts which are bent respectively to form a square of side=x units and a circle of radius=r units. If the sum of the areas of the square and the circle so formed is minimum, then :

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side=x units and a circle of radius=r units.  If the sum of the areas of the square and the circle so formed is minimum, then :

• Option 1)

2x = (π+4) r

• Option 2)

(4−π) x = πr

• Option 3)

x = 2r

• Option 4)

2x = r

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As we learnt in

Circle -

A circle is the locus of a moving point such that its distance from a fixed point is constant.

- wherein

Let the length of two parts be 'a' and '2-a'

$a=4x$  and  $2-a=2\pi r$

where x is side of square

sum of Areas $= \frac{a^2}{16}+\frac{a^2-4a+4}{4\pi }$

of $\left ( a \right ) = \frac{a^2\pi +4a^2-16a+16}{16\pi }$

$f^{'}\left ( a \right ) = \frac{1}{16\pi }\left [ 2a\pi +8a-16 \right ]=0$

$2a\pi +8a=16$

$x= \frac{a}{4} = \frac{2}{\pi +4}$

$r=\frac{2-a}{2\pi }$

$r=\frac{1}{\pi +4}$

Hence   $x=2r$

Option 1)

2x = (π+4) r

Incorrect

Option 2)

(4−π) x = πr

Incorrect

Option 3)

x = 2r

Correct

Option 4)

2x = r

Incorrect

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