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Math Help - ladder with angle

  1. #1
    Junior Member
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    ladder with angle

    A 12 foot solar panel is to be installed on a roof with a 15 degree pitch. Find the length of the vertical brace d if the panel must be installed to make a 40 degree angle with the horizontal.

    I guess what i really don't understand if where the 40 degree angle is....
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  2. #2
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    Hello, kelsey3!

    A 12-foot solar panel is to be installed on a roof with a 15 pitch.
    Find the length of the vertical brace d if the panel
    must make a 40 angle with the horizontal.
    Code:
                                  * B
                              *   |d
                     12   *       * C
                      *     *     |
                  *   *           |
              * * 15             |
        A * - - - - - - - - - - - * D

    The solar panel is: . AB = 12
    \angle CAD = 15^o,\;\angle BAD = 40^o

    In right triangle BDA\!:\;\begin{array}{cccccccc}\sin40^o &=& \frac{BD}{12} &\Rightarrow&  BD &=& 12\sin40^o& {\color{blue}[1]}\\ \\[-4mm] <br />
\cos40^o &=& \frac{AD}{12} & \Rightarrow & AD & =& 12\cos40^o & {\color{blue}[2]}\end{array}

    In right triangle CDA\!:\;\;\tan15^o \,=\,\frac{CD}{AD}\quad\Rightarrow\quad CD \:=\:AD\tan15^o \;{\color{blue}[3]}

    Substitute [2] into [3]: . CD \:=\:12\cos40^o\tan15^o\;{\color{blue}[4]}


    We have: . d \;=\;BC \;=\;BD - CD\;{\color{blue}[5]}


    Substitute [1] and [4] into [5]: . \boxed{d \;=\;12\sin40^o - 12\cos40^o\tan15^o}

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