Question

Only smart,intelligent,aggressive,modern,iitian can solve this

1. 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
(1) (4-π)x = πr

(2) x = 2r

(1) 2x = r

(2) 2x = (π + 4)r

pls answer as fast as u can

Answer 1

Let length of two parts be ‘a’ and ‘2 - a’
As per condition given,
a=4xand2−a=2πra=4xand2−a=2πr
x=a4x=a4 and r=2−a2πr=2−a2π
∴A(square)=(a4)2=a216∴A(square)=(a4)2=a216 and
A(circle)=π[(2−a)2π]2=π(4+a2−4a)4π2A(circle)=π[(2−a)2π]2=π(4+a2−4a)4π2
=(a2−4a+4)4π=(a2−4a+4)4π
f(a)=a216+a2−4a+44πf(a)=a216+a2−4a+44π
∴f(a)=a2π+4a2−16a+1616π∴f(a)=a2π+4a2−16a+1616π
f′(a)=116πf′(a)=116π[2aπ+8a−16][2aπ+8a−16]
f′(a)=0=>2aπ+8a−16=0f′(a)=0=>2aπ+8a−16=0
=> 2aπ+8a=162aπ+8a=16
x=a4=2π+4x=a4=2π+4
and r=2−a2πr=2−a2π
=2−8π+42π=2−8π+42π
=2π+8−82π(π+4)=2π+8−82π(π+4)
=1π+4=1π+4
x=2π+4x=2π+4 and r=1π+4r=1π+4
◆x=2r

(2) x = 2r is the correct answer.

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