Question 985053
A flowerbed is made in shape of sector of circle, 20 meter of wire is available to make a fence for the flowerbed ,
find the radius of circle so that the area of flowerbed is maximum? 
:
let r = the radius
let a = length of the arc of the sector
then
2r + a = 20
a = (20-2r)
the area of the sector; 
The area of the circle times the fraction of the circumference that is the arc
A = {{{pi*r^2}}} * {{{a/(2*pi*r)}}}
replace a
A = {{{pi*r^2}}} * {{{(20-2r)/(2*pi*r)}}}
simplify, cancel pi and r in the denominator
A = r*{{{(20-2r)/2}}}
Cancel 2
A = r(10-r)
This is a simple quadratic equation now
A = -r^2 +10r
Max area occurs at the axis of symmetry we can find using x=-b/(2a)
here a=-1; b=10, x=r
r = {{{(-10)/(2*-1)}}}
r = -10/-2
r = 5 meters is the radius
:
:
A graph of this equation, r on the x axis, area on the y axis
{{{ graph( 300, 200, -6, 12, -10, 30, x*(10-x)) }}}
Max area occurs at r = 5 meters, max area is 25 sq meters