A 36 inches piece of string is cut into two pieces. one piece is used to form a circle while the other is used to form a circle. how should the string be cut so that the sum of the area is a minimum.
those are the circumferences so x/(2pi)=radious of the x circle (36-x)/2pi=radius of the y circle
the area of each is the area of the x circle will be x²/(4pi) the area of the y circle will be (36-x)²/(4pi) or (x²-72x+1296)/(4pi)
the sum of the areas is (2x²-72x+1296)/(4pi) or (x²-36x+648)/(pi) find the minimum value basically find the value of x that makes it minimum take derivitive dy/dx=pi(2x-36) set equal to 0 0=pi(2x-36) 0=2x-36 36=2x x=18 at x=18, the derivitive changes from negative to positive so the minimum occurs at x=18 y=36-x=36-18=18
so the string should be cut in half the areas would be about 51.5
