Question 744782
You have a wire that is 53 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? 
:
let r = the radius of the circle
let s = the side of the square
:
We know the perimeter of the square and the circumference = 53, therefore
4s + {{{2pi*r}}} = 53
therefore
4s = 53 - {{{2pi*r}}}
s = {{{(53-(2pi*r))/4}}}
:
Total area
A = {{{pi*r^2}}} + {{{s^2}}}
:
Replace s with {{{(53-(2pi*r))/4}}}
A = {{{(pi*r^2)}}}+{{{((53-(2pi*r))/4)^2}}}
Find the minimum area by graphing this equation
{{{ graph( 300, 200, -4, 10, -20, 200, ((pi*x^2)+((53-(2*pi*x))/4)^2)) }}}
Putting the same equation in my Ti83, minimum area occurs when r=3.71
therefore
C = {{{2*pi*3.71}}}
C = 23.3 cm, circumference of the circle for min area of the circle & square