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Math Help - Optimization: Minimize Paper

  1. #1
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    Optimization: Minimize Paper

    The upper right-hand corner of a piece of paper, 12 in by 8 in, as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y?
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  2. #2
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    Re: Optimization: Minimize Paper

    Quote Originally Posted by AXQ4286 View Post
    The upper right-hand corner of a piece of paper, 12 in by 8 in, as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y?
    I don't know what you have done so far and where you are stuck ... So here are some hints:

    1. I've modified you sketch a little bi( see attachment)

    2. The green-bordered triangle is congruent to the blue-bordered triangle.
    3. x must be longer than 4.
    4. The two colour-bordered triangles form a rhombus whose area is

    \frac12 \cdot y \cdot f = 2 \cdot \frac12 \cdot x \cdot k

    5. You are dealing with a lot of right triangles. For instance:

    • h^2+(8-x)^2=x^2
    • h^2+8^2=f^2
    • x^2+k^2=y^2
    • (k-h)^2+8^2=k^2


    ... and now it's your turn.
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  3. #3
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    Re: Optimization: Minimize Paper

    Quote Originally Posted by AXQ4286 View Post
    The upper right-hand corner of a piece of paper, 12 in by 8 in, as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y?
    I've constructed some of all line segments y. The radius of the circles is x.

    You can use this drawing to confirm the result of your calculations. (Btw: The minimum of y is not drawn!)
    Attached Thumbnails Attached Thumbnails Optimization: Minimize Paper-kurzfaltprozess.png  
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