1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Volume of a solid

  1. Jul 4, 2003 #1
    Hi,
    I need help on this problem which is giving me a few headaches...!!!!

    here goes..

    Show that the volume of the solid bounded by the coordinate planes and the plane tangent to the portion of the surface xyz = k, k>0, in the first octant does not depend on the point of tangency.

    Your help will be much appreciated.
     
  2. jcsd
  3. Jul 4, 2003 #2
    OK, i think i have the answer. i make it 9k/2. how much help do you want?

    my first hint: the normal to that surface can be found by taking the gradient.
     
  4. Jul 4, 2003 #3
    Hi lethe,

    I am totally lost on this question to be honest and I can't seem to work out what to do here to solve it. I would really appreciate it if you could explain step by step what you are doing so I can understand how you came to your conclusion and your answer.

    eg. how you came to your answer of 9k/2.

    and also your hint: the normal to that surface can be found by taking the gradient.

    How would you go about solving this?

    Your help will be greatly appreciated.
    Thanks.
     
  5. Jul 4, 2003 #4
    OK:

    step 1: the gradient of xyz - k gives you the normal vector to the surface.

    the gradient is (yz,xz,xy)

    step 2: the equation for a plane with normal vector n is n*(x-x0)=0

    so the equation for the tangent plane at x0 is y0z0(x-x0)+x0z0(y-y0)+x0y0(z-z0)=0

    or

    x/x0 + y/y0 + z/z0 = 3

    step 3: find the three coordinate intercepts of this plane by plugging in x=y=0 and get z=30, then x=z=0 and get y=3y0, and x=3x0

    step 4: calculate the volume. it is a right pyramid, the base has legs 3x0 and 3y0, so the area of the base is 9x0y0/2. the area for a pyramid is 1/3*Base*height, so this is 9x0y0z0/2, but since x0 is on the surface, x0y0z0 = k, and we get 9k/2 for the volume
     
  6. Jul 5, 2003 #5
    Hey thanks for your help lethe!

    Regards
    Iceman
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Volume of a solid
  1. Volume of a solid (Replies: 1)

  2. Solid angle (Replies: 7)

  3. Imaginary volume (Replies: 1)

Loading...