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27 October, 18:56

You have a wire that is 29 cm long. you wish to cut it into two pieces. one piece will be bent into the shape of a square. the other piece will be bent into the shape of a circle. let a represent the total area of the square and the circle. what is the circumference of the circle when a is a minimum?

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  1. 27 October, 21:35
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    C = circumference of circle

    p = perimeter of square

    c + p = 29

    p = 29 - c

    side of square = (29 - c) / 4

    radius of circle = c / (2pi)

    a = side^2 + (pi) r^2

    a = ((29 - c) / 4) ^2 + pi * (c / (2pi)) ^2

    a = (1/16) (841 - 58c + c^2) + (c^2 / (4pi^2)) * pi

    a = 52.5625 - 3.625c + c^2/16 + c^2 / (4pi)

    Since the 2nd degree term has a positive coefficient is positive, this has a graph that is a parabola opening up. It has a minimum value.

    We take the derivative of the area expression and set equal to zero to find the minimum circumference.

    da/dc = - 3.625 + c/8 + (c / (2pi)) = 0

    c = 12.76

    Answer: 12.76 cm
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