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Math Help - Minimizing Area

  1. #1
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    Post Minimizing Area

    A 36-in. piece of string is cut into two pieces. One piece is used to form a circle while the other is used to form a square. How should the string be cut so that the sum of the areas is a minimum? Help!

    Thank you!
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  2. #2
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    one piece has length x ... make that length into a circle

    other piece has length (36-x) ... make that length into a square


    circumference, C = x = 2\pi r

    r = \frac{x}{2\pi}

    circle area, \pi r^2 = \pi \frac{x^2}{4\pi^2} = \frac{x^2}{4\pi}

    side of the square, s = \frac{36-x}{4}

    square area, s^2 = \frac{(36-x)^2}{16}

    total area ...

    A = \frac{x^2}{4\pi} + \frac{(36-x)^2}{16}
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