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Math Help - calc optimization problem

  1. #1
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    calc optimization problem

    The 6 segment ofthewindow frame shown in the diagram are tobe constructed from a piece of window framing material 6 m in length. A carpenterwants to build a frame for a rural gothic style window, where triangle ABC is equilateral .The window must fit inside a space that is 1 m wide and 3 m high.
    Please help me if you can . Thank you in advance!

    sorry my bad i forgot to post the rest of the question and the diagram which is basically a triangle on top of the square like this
    .................................................. .............................A
    .................................................. .............................*....
    .................................................. ......................B *.......* C
    .................................................. ....................... *******
    .................................................. ........................*.........*
    .................................................. ........................*.........*
    .................................................. ........................*******

    Determine the dimensions that should be used for the 6 pieces so that the maximum amount of light will be admitted.Assume no waste material for corner cuts and so on.
    Last edited by anna12345; April 8th 2009 at 02:40 PM.
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  2. #2
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    Quote Originally Posted by anna12345 View Post
    The 6 segment ofthewindow frame shown in the diagram ...
    diagram?
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  3. #3
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    Hello, Anna!

    The 6 segments of the window frame shown in the diagram
    are to be constructed from a piece of window framing material 6 m in length.
    A carpenter wants to build a frame for a rural gothic style window,
    where triangle ABC is equilateral.

    The window must fit inside a space that is 1 m wide and 3 m high. . . . . not needed
    Determine the dimensions that will maximize the area of the window.
    Code:
              B
              *
             / \
          x /   \ x
           /     \
        A * - - - * C
          |   x   |
        y |       | y
          |       |
          * - - - *
              x
    The total length of the framework is . 4x + 2y \:=\:6 \quad\Rightarrow\quad y \:=\:3 - 2x\;\;{\color{blue}[1]}

    The area of an equilateral triangle of side x is: . \tfrac{\sqrt{3}}{4}x^2
    The area of the rectangle is: . xy

    The total area is: . A \;=\;\tfrac{\sqrt{3}}{4}x^2 + xy\;\;{\color{blue}[2]}


    Substitute [1] into [2]: . A \;=\;\frac{\sqrt{3}}{4}x^2 + x(3-2x)

    . . and that is the function you must maxiimize.

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