SOLUTION: A flowerbed is made in shape of sector of circle, 20meter of wire is available to make a fence for the flowerbed ,find the radius of circle so that the area of flowerbed is maximum

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Question 985053: A flowerbed is made in shape of sector of circle, 20meter of wire is available to make a fence for the flowerbed ,find the radius of circle so that the area of flowerbed is maximum?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
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%2Ar%5E2 * a%2F%282%2Api%2Ar%29
replace a
A = pi%2Ar%5E2 * %2820-2r%29%2F%282%2Api%2Ar%29
simplify, cancel pi and r in the denominator
A = r*%2820-2r%29%2F2
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 = %28-10%29%2F%282%2A-1%29
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%28+300%2C+200%2C+-6%2C+12%2C+-10%2C+30%2C+x%2A%2810-x%29%29+
Max area occurs at r = 5 meters, max area is 25 sq meters