Question

You have a wire that is 38 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?

Give your answer to two decimal places

Answer 1

Answer:16.71 cm

Explanation:

Given

Length of wire L=38 cm

One piece is bent in the form of square and another in the form of circle

let x be the length of circle

therefore length of square side
`(38-x)/(4)`

A=total area of square and circle

radius of circle
`r=(x)/(2\pi )`

area of circle
`A_c=\pi r^2=\pi * ((x)/(2\pi ))^2`

Area of square
`A_s=((38-x)/(4))^2`

`A=\pi * ((x)/(2\pi ))^2+((38-x)/(4))^2`

To get the minimum value of A we get

`\frac{\mathrm{d} A}{\mathrm{d} x}=(2x)/(4\pi )-(2(38-x))/(16)`

`\frac{\mathrm{d} A}{\mathrm{d} x}=0`

`(x)/(4\pi )=(38-x)/(16)`

`x=(38\pi )/(4+\pi )`

Therefore circumference of circle

`x=(38\pi )/(4+\pi )=(119.396)/(7.142)=16.717 cm`