SOLUTION: 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 ca

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Question 939075: 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.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
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.
:
We know that the circumference of the circle + the perimeter of the square = 40
Let x = the circumference of the circle
Find r in terms of x
2%2Api%2Ar+=+x
r = x%2F%282%2Api%29
Area+=+pi%2Ar%5E2,
therefore
A = pi%2A%28x%2F%282%2Api%29%29%5E2 = pi%2A%28x%5E2%2F%284%2Api%5E2%29%29 = x%5E2%2F%284%2Api%29 is the area of the circle
:
(40-x) = the perimeter of the square
then
%28%2840-x%29%29%2F4 = one side of the square
and
%28%28%2840-x%29%29%2F4%29%5E2 = the area of the square
:
Total Area
A(x) = x%5E2%2F%284%2Api%29 + %28%28%2840-x%29%29%2F4%29%5E2
Graph this equation

Put this equation into my graphing calc, min occurs when x = 17.6 cm is the piece to make the circle
therefore the piece of wire to make the square is 40 - 17.6 = 22.4 cm
;
Did this make sense to you? let me know; ankor@att.net