Question 842373
You have a wire that is 65 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? 
:
perimeter + circumference = 65
let {{{2pi*x)}}} = circumference of the circle
then
{{{(65-(2*pi*x))}}} = perimeter of the square
:
{{{(65-(2*pi*x))/4}}} = one side of the square
and
{{{((65-(2*pi*x))/4)^2}}} = the area of the square
Find the area of the circle
a = {{{pi*x^2}}}
Find the total area
A = {{{((65-(2*pi*x))/4)^2}}} + {{{pi*x^2}}}
Graphically
{{{ graph( 300, 200, -3, 10, -100, 300, ((65-(2*pi*x))/4)^2+(pi*x^2))) }}}
minimum area occurs when x = 4.55
C = {{{2*pi*4.55}}}
 C = 28.6 is the circumference with minimum area