Question

You have a wire that is 44 cm long. You wish to cut it into two pieces. One piece will be bent into the shape of a square. The other piece will be bent into the shape of a circle. Let A represent the total area of the square and the circle. What is the circumference of the circle when A is a minimum?

Answer 1

The circumference of the circle is 29.36 cm

Explanation:

Let

x -----> the length of first piece (shape of square)

y ------> the length of the other piece (shape of a circle)

we know that

`x+y=44`

`y=44-x` -----> equation A

step 1

Find out the area of square

The perimeter of square is equal to the length of the first piece

`P=4b`

where

b is the length side of square

`P=x\ cm`

`x=4b`

`b=x/4`

Find the total area A

The area of square is

`A_1=b^2`

`A_1=(x^(2))/(16)`

step 2

Find out the area of the circle

The circumference of the circle is equal to the length of the other piece

`C=(44-x)\ cm`

The circumference is equal to

`C=2\pi r`

so

`2\pi r=(44-x)\ cm`

Find the radius of the circle

`r=((44-x))/(2\pi)\ cm`

Find the area of the circle

`A_2=\pi r^(2)`

substitute the value of r

`A_2=\pi (((44-x))/(2\pi))^(2)`

`A_2= ((44-x)^2)/(4\pi)`

step 3

Find out the total area

`A=A_1+A_2`

substitute

`A=(x^(2))/(16)+((44-x)^2)/(4\pi)`

This is a vertical parabola open upward

The vertex is a minimum

using a graphing tool

The vertex is the point (14.64, 67.77)

see the attached figure

For x=14.64 cm -----> the area is a minimum

The lengths of the wire are

`x=14.64\ cm\\y=44-14.64=29.36\ cm`

therefore

The circumference of the circle is 29.36 cm