Question 503532
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 x = circumference of the circle
then
(53-x) = the perimeter of the square
and
{{{(53-x)/4}}} = the side of the square
and
{{{((53-x)/4)^2}}} = the area of the square
:
Find the area of the circle using the circumference
find the radius (r)
2*pi*r = x
r = {{{x/(2*pi)}}}
r = {{{x/6.28}}}
Find the area of the circle
A = {{{pi*r^2}}}
Replace r with {{{x/6.28}}}
A = {{{pi}}}*{{{(x/6.28)^2}}} =  {{{pi}}}*{{{(x^2/39.48)}}}
cancel pi into 39.48
A = {{{x^2/12.566}}}
Total area of circle and square
A = {{{x^2/12.566}}} + {{{((53-x)/4)^2}}} = {{{x^2/12.566}}} + {{{((2809-106x+x^2)/16)}}}
:
convert these fractions to decimal coefficients
A(x) = .0796x^2 + .0625x^2 - 6.625x + 175.5625
A(x) = .1421x^2 - 6.625x + 175.5625
:
Find the axis of symmetry of this quadratic equation (min area)
x = {{{(-(-6.625))/(2*.1421)}}}
x = {{{6.625/.2842}}}
x = 23.31 cm is the circumference when they have min area