Question

A rope which is x m long is cut into 5 equal parts.
What is length of each part?
If the total length is 50cm,
what is the length of 2 parts?​

Answer 1

Length of wire =50 m (Given)

Let length of one piece for shape of square =x m

∴ Length of other piece for shape of circle

=(50−x)m

Now perimeter of square =4a=x

⇒a=4x

and circumference of circle =2πr=50−x

⇒r=2π50−x

Combined Area =a2+πr2

=16x2+π⋅(2π50−x)2

=16x2+π⋅4π2(50−x)2

A=16x2+4π(50−x)2

Differentiating w.r. to x, we get

dxdA=162x+4π2(50−x)(−1)

dxdA=8x+2π(x−50)

=8ππx+4x−200

=8πx(4+π)−200

For extremum, dxdA=0

∴x(4+π)−200=0

x=4+π200

dx2d2A>0

A is minimum at x=4+π200

∴ Length of square of square wire, x=4+π200m

and length of circle wire =50−x

=50−4+π200