SOLUTION: A piece of wire 40cm long is to be cut into two pieces. One piece wil be bent to form a circle; the other will be bent to form a square. Find the lengths of the two pieces that ca

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Question 263596: A piece of wire 40cm long is to be cut into two pieces. One piece wil be bent to form a circle; the other will be bent to form a square. Find the lengths of the two pieces that cause the sum of the area of the circle and square to be a minimum.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A piece of wire 40cm long is to be cut into two pieces.
One piece will be bent to form a circle; the other will be bent to form a square.
Find the lengths of the two pieces that cause the sum of the area of the circle and square to be a minimum.
:
Let x = circumference of the circle
then
(40-x) = perimeter of the square
:
Find the radius of the circle
r = x%2F%282pi%29
Find the area of the circle
a = pi%28x%2F%282pi%29%29%5E2+
a = pi%28x%5E2%2F%284pi%5E2%29%29
Cancel pi
a = x%5E2%2F%284pi%29
:
Find the area of the square:
a = %28%2840-x%29%2F4%29%5E2
a = %28x%5E2+-+80x+%2B+1600%29%2F16
:
:
Total area
A = x%5E2%2F%284pi%29 + %28x%5E2+-+80x+%2B+1600%29%2F16
Change to decimals, easier to combine like terms
A = .07958x^2 + .0625x^2 - 5x + 100
A = .14208x^2 - 5x + 100
:
Find the axis of symmetry [x = -b/(2a)]
x = %28-%28-5%29%29%2F%282%2A.14208%29
x = 5%2F.28416
x ~ 17.6 inches, the piece of wire creating a circle
and
40 - 17.6 = 22.4 inches, the piece of wire creating the square
:
These lengths should give minimum area of the circle and square together