A36 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.

Well, ok

the amounts cut off are x and y
x=x and y=36-x

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