Question

A wire of length l is cut into two parts. One part is bent into a circle and other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half the side of the square.

Answer 1

Let length of two parts be ‘a’ and ‘2 - a’

As per condition given, we write

a=4xand2−a=2πr

x=a4 and r=2−a2π

∴A(square)=(a4)2=a216 and

A(circle)=π[(2−a)2π]2=π(4+a2−4a)4π2

=(a2−4a+4)4π

f(a)=a216+a2−4a+44π

∴f(a)=a2π+4a2−16a+1616π

f′(a)=116π[2aπ+8a−16]

f′(a)=0=>2aπ+8a−16=0

=> 2aπ+8a=16

x=a4=2π+4

and r=2−a2π

=2−8π+42π

=2π+8−82π(π+4)

=1π+4

x=2π+4 and r=1π+4

x=2r