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Math Help - Surfaces

  1. #1
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    Surfaces

    Consider the following surface:

    x = 4y^2 + 4z^2

    What's this surface called?

    What coordinate plane does z=0 define?

    What's the trace of this surface in the xz plane look like?

    What's the trace of this surface in the planes x=k look like?
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  2. #2
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    Quote Originally Posted by Ideasman View Post
    Consider the following surface:

    x = 4y^2 + 4z^2

    What's this surface called?
    What coordinate plane does z=0 define?
    What's the trace of this surface in the xz plane look like?
    What's the trace of this surface in the planes x=k look like?
    Hello,

    I've attached a diagram of this surface.

    a) It is a rotation paraboloid. The axis of rotation is the positive x-axis.

    b) It's a parabola with the vertex in the origin: x = 4y^2

    c) It's a circle: k = 4x^2+4y^2 \Longleftrightarrow x^2+y^2=\left( \frac{1}{2} \sqrt{k} \right)^2

    EB
    Attached Thumbnails Attached Thumbnails Surfaces-rotat_paraboloid1.gif  
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  3. #3
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    Hello, Ideasman!

    I use the standard orientation of the three coordinate axes.
    Code:
                      z
                      |
                      |
                      |
                      |
                      * - - - - - - y
                    /
                  /
                /
              x
    Consider the following surface: . x \:= \:4y^2 + 4z^2

    What's this surface called?
    It is a paraboloid.


    What coordinate plane does z=0 define?
    z = 0 is the xy-plane (the "floor" of the graph).


    What's the trace of this surface in the xz-plane look like?
    The xz-plane is the "left wall" of the graph.

    Let y = 0 and we have: . x = 4z^2

    This is a parabola on the "left wall", vertex at the origin,
    . . opening in the positive x-direction.



    What's the trace of this surface in the planes x=k look like?

    Let x = k. .We have: . 4y^2 + 4z^2\:=\:k\quad\Rightarrow\quad y^2 + z^2\:=\:\frac{k}{4}

    These are circles: centered on the x-axis with radius \frac{\sqrt{k}}{2}

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