A piece of wire 20 cm long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. Atwhat length should the wire be cut to minimize the total area enclosed?...

A piece of wire 20 cm long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. At
what length should the wire be cut to minimize the total area enclosed?

kindly help me in solving this problem?

Asked on by hinaje

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aruv | High School Teacher | (Level 2) Valedictorian

Posted on

Let length of one piece of wire be x, which bent into circle.

Thus circumference of cirle be x. Let radius of the circle be r, so

`2pir=x`

`r=x/(2pi)`

Thus area of the circle be

`A_c=pi(x/(2pi))^2=x^2/(4pi)`              (i)

Perimeter of the square be

P=20-x

Thus side of the square be (20-x)/4

Thus area of the square be

`A_s=(5-x/4)^2`

Total enclosed area be

`A=A_c+A_s`

`A=x^2/(4pi)+(5-x/4)^2`

`(dA)/(dx)=x/(2pi)-2(5-x/4)(1/4)`

`=x/(2pi)-5/2+x/8`

For maximum / minimum, we have

`((4+pi)x)/(8pi)-5/2=0`

`x=(5/2)((8pi)/(4+pi))`

`x=(20pi)/(4+pi)`                (ii)

`(d^2A)/(dx^2)=1/(2pi)+1/8>0AAx.`

`Thus`

x=8.798 cm.

Thus pieces of wire should be  11.202 cm. and 8.798 cm.

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