Results 1 to 3 of 3

Thread: Surfaces

  1. #1
    Member
    Joined
    Sep 2006
    Posts
    221

    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?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,854
    Thanks
    138
    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: $\displaystyle x = 4y^2$

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

    EB
    Attached Thumbnails Attached Thumbnails Surfaces-rotat_paraboloid1.gif  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    12,028
    Thanks
    848
    Hello, Ideasman!

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

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


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


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

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

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



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

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

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

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Intersection of surfaces
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Nov 20th 2011, 03:38 AM
  2. Surfaces
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 28th 2010, 11:31 AM
  3. surfaces
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Apr 6th 2010, 11:24 AM
  4. help with quadric surfaces
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Sep 3rd 2008, 12:15 AM
  5. Parametric surfaces
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jun 24th 2008, 10:32 AM

Search Tags


/mathhelpforum @mathhelpforum