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Math Help - Applications of vectors and trigonometry to 3d cubes... Really desperate for help

  1. #1
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    Applications of vectors and trigonometry to 3d cubes... Really desperate for help

    Hi. So I need help with some math questions. I've asked my friends but they're kind of stuck too.

    1.http://s1295.photobucket.com/user/chloe ... 5.png.html
    It is supposed to be a cube that is sticking out of the ground. We were given:
    The ground coincides with the x,y-plane and that the y-axis is supposed to point West and the x-axis North
    A = (12.9258, 31.0732, 8.8878) and C = (2.1003, 4.5653, 17.4866)
    B and D are at equal height above the ground
    And all these numbers are measurements in meters

    So I've managed to find the other points:
    A=(12.9258,31.0732,8.8878)
    B=(-6.9962,21.4153,13.1872)
    C=(2.1003,4.5653,17.4866)
    D=(22.3976,19.1976,13.1872)

    A'=(12.6675,27.6495,0)
    B'=(-7.3795,16.3354,0)
    C'=(1.5921,-2.1709,0)
    D'=(22.0143,14.1177,0)

    |AA'|=9.5279m
    |BB'|=14.1370m
    |CC'|=18.7461m
    |DD'|=14.1370m
    |AB|=22.5532m

    〖Area〗_A'B'C'D'=378.5771m^2

    But I still cannot figure out how to determine the volume of the cube above and below the ground. At first i thought the volume of cube above was split into two parts, a rectangle and a triangle, but then I thought it could be a trapezoid? I've tried using both formulas, but I still cant seem to wrap my head around it. I've found the volume of the total cube to be 11471.54017m^3. Using the formula for the volume of a trapezoid i ended up with a value much larger than the volume i got for the cube itself!

    2.http://s1295.photobucket.com/user/chloe ... 9.png.html

    This is another cube (Cube2) that has the same dimensions as the one above. It will be sunk into the ground such that the center of the cube is at ground level and such that one of the four diagonals of Cube2 is at right angles to the ground.

    If the sun is right above, what shape is the shadow that Cube2 casts? Sketch this shadow.

    So I found out that where the water intersects it creates a hexogonal shape, and my friends say its supposed to be a nonagonal shape shadow but I don't see it.

    Thanks you so much for helping!
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  2. #2
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    Re: Applications of vectors and trigonometry to 3d cubes... Really desperate for help

    Hey cchloe93.

    Hint: Solve for the intersection of the cube and the ground and then break everything up into triangular pyramid solids and add the volumes together.
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  3. #3
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    Re: Applications of vectors and trigonometry to 3d cubes... Really desperate for help

    Hey, i'm sorry but I dont really understand the hint?
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  4. #4
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    Re: Applications of vectors and trigonometry to 3d cubes... Really desperate for help

    Basically what you do is divide the cube into the part of it above the ground with the part below. You'll get some complex shapes for both in general.

    Then you break everything up into triangular solids and calculate the volume for each solid to get the volume for all solids above or below ground.

    Also I should have mentioned that the intersection points can be found by finding the intersection of all edges with the ground plane.
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