Question

A piece of wire 20 ft. long is cut into two pieces. One piece is made into a circle and the other piece is made into a square. Let the piece of length x be formed into a circle. How long should each piece of wire be to minimize the total​ area? What is the radius of the​ circle? How long is each side of the​ square?

Answer 1

Radius of circle =
`(10)/(4+\pi)`

Length of wire which is made into circle=
`(20\pi)/(4+\pi)`

Length of wire which is made into square =
`(80)/(4+\pi)`

Explanation:

We are given that a piece of wire 20 ft and cut into two pieces.

One piece is made into a circle and other piece is made into a square.

We have to find the length of each piece of wire when total area is minimum and find the value of radius.We have to find the length of each side of square.

Let r be radius of circle and y be the side of square

Total length of wire=Circumference of circle + perimeter of circle

`20=2\pi r+4y`

`y=(10-\pi r)/(2)`

Total area =Area of circle +Area of square

`A=\pi r^2+y^2`

`A=\pi r^2+((10-\pi r)/(2))^2`

Differentiate w.r.radius

`(dA)/(dr)=2\pi r-(\pi)/(2)(10-\pi r)`

Substitute
`(dA)/(dr)=0`

`4\pi r-10 \pi+\pi^2 r=0`

`4r-10+\pi r=0`

`r(4+\pi)=10`

`r=(10)/(4+\pi)`

Differentiate w.r.t radius

`(d^2A)/(dr^2)=2\pi+(\pi^2)/(2)` > 0

Hence, the total area is minimum for
`r=(10)/(4+\pi)`

Substitute the values then we get

Length of wire which is made into circle=
`(20\pi)/(4+\pi)`

Length of wire which is made into square =
`20-(20\pi)/(4+\pi)`

Length of wire which is made into square =[tex]\frac{80}{4+\pi}[\tex]