1. Not finding help here? Sign up for a free 30min 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!

Multivariable Optimization Problem

  1. Aug 10, 2004 #1
    I have two questions.

    A) Show the parallelipided with fixed surface area and maximum volume is a cube.

    I've already proven that we can narrow down the proof to a box. So, basically, I'm really lost on how do prove that a cube is the box with a fixed surface area and maximum volume.

    B) We might not have covered how to do part B yet, so i'll create a new topic if I still don't understand after tomorrow's lecture.
     
  2. jcsd
  3. Aug 10, 2004 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    How is a "box" different from a parallelpiped?

    Call the lengths of the sides of your parallelpiped x, y, and z.

    The volume is V= xyz.

    The surface area is 2xy+ 2xz+ 2yz= A (a constant).

    Now use the "Lagrange multiplier" method.

    In order that V= xyz be a minimum (or maximum!) on the surface U=2xy+2xz+ 2yz- A=0, the two gradient vectors, grad V= <yz, xz, xy> and grad U= <2y+ 2z,2x+ 2z, 2x+ 2y> must be parallel. That is we must have <yz, xz, xy>= some multiple of <2y+2z, 2x+ 2z, 2x+ 2y> so that yz= &lambda;(2y+ 2z), xz= &lambda;(2x+ 2z), and
    xy= &lambda;<2x+ 2y>. Eliminate &lambda; from tose equations and see what happens.
     
  4. Aug 11, 2004 #3

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    A box is characterized with right angles, whereas a parallellepiped need not be subject to this constraint.
     
  5. Aug 11, 2004 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Ah, right. Thanks.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Multivariable Optimization Problem
  1. Optimization problem (Replies: 1)

  2. Optimization Problems. (Replies: 3)

  3. Optimizing problem (Replies: 2)

Loading...