SOLUTION: 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

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Question 543147: 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
Answer by lwsshak3(11628) (Show Source):
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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.
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let x=length of first piece of string
36-x=length of second piece of string
..
x=2πr=circumference of first circle
r=x/2π
area=πr^2=π*(x/2π)^2=πx^2/4π^2=x^2/4π
..
36-x=2πr=circumference of second circle circle
r=36-x/2π
area=πr^2=π*(36-x/2π)^2=π(36-x^2/4π^2=(36-x)^2/4π
..
Add areas:
x^2/4π+(36-x)^2/4π
x^2/4π+(1296-72x+x^2)/4π
(2x^2-72x+1296)/4π
differentiate:
f'(x)=(4x-72)/4π=0
4x-72=0
4x=72
x=18
..
f"(x)=4/4π=(1/π)>0
Therefore, 18 is x min
ans:
The string should be cut exactly in half with each piece 18 inches in length.