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Math Help - maximize and minimize

  1. #1
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    maximize and minimize

    A wire 20 m long is cut into two pieces. One piece is bent in the shape of
    an equilateral triangle, and the other is bent in the shape of a circle. How
    should the wire be cut to maximize the total area enclosed by the shapes?
    How should it be cut to minimize the total area?

    any help?
    i got complicated in the numbers :S
    root 3 and pi !!

    regards
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  2. #2
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    Quote Originally Posted by Abbas View Post
    A wire 20 m long is cut into two pieces. One piece is bent in the shape of
    an equilateral triangle, and the other is bent in the shape of a circle. How
    should the wire be cut to maximize the total area enclosed by the shapes?
    How should it be cut to minimize the total area?

    any help?
    i got complicated in the numbers :S
    root 3 and pi !!

    regards
    x = perimeter of triangle ... each side s = \frac{x}{3}

    area of an equilateral triangle ...

    A = \frac{\sqrt{3}}{4}s^2 = \frac{\sqrt{3}}{4}\left(\frac{x}{3}\right)^2


    20-x = circumference of circle

    2\pi r = 20-x

    r = \frac{20-x}{2\pi}

    A = \pi\left(\frac{20-x}{2\pi}\right)^2

    total area ...

    A = \pi\left(\frac{20-x}{2\pi}\right)^2 + \frac{\sqrt{3}}{4}\left(\frac{x}{3}\right)^2

    find \frac{dA}{dx} and optimize
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  3. #3
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    u'll get one value for x only , so how it will be cut to maximize if the value was local minimum ,?
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  4. #4
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    Quote Originally Posted by Abbas View Post
    u'll get one value for x only , so how it will be cut to maximize if the value was local minimum ,?
    consider the endpoint extrema.
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  5. #5
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    what the domain will be?
    is it from [0,20]?
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  6. #6
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    Quote Originally Posted by Abbas View Post
    what the domain will be?
    is it from [0,20]?
    yes ...

    x = 0 , all circle

    x = 20 , all triangle
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