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.

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