What are Lagrange Multipliers?

Date: 02/09/2000 at 02:09:36
From: Melissa Martin
Subject: What are Lagrange Multipliers?

Dear Dr. Math,

I've just received an independent assignment for my Calculus class and 
I don't understand any of it. It's all about Lagrange Multipliers - 
what exactly is a Lagrange Multiplier? How do they work and what 
method can I use to solve the problems I have been given on this 
assignment? One of my questions is:

Squigets are stored in a rectangular box with no top and with volume 
2500 cc. Three different materials are used in the construction of 
this box. The bottom is made out of material that costs 5 cents/cm, 
the front and back are made out of material that costs 4 cents/cm, and 
material for the two sides costs 2 cents/cm. What are the dimensions 
of the box (with volume 2500) that will minimize the total cost of the 
materials? Note here that you will have 3 variables, the width of the 
box X, the depth Y and the height Z, so you will have 3 multipliers, 
which at the optimum must all be equal. Also the constraint is not 
linear.

Please help. Thank you,
Melissa

Date: 02/09/2000 at 10:42:49
From: Doctor Anthony
Subject: Re: What are Lagrange Multipliers?

I will start with a general note on Lagrange multipliers since you say 
that you are not familiar with the concept. See:

  http://mathforum.org/dr.math/problems/aleja1.8.98.html   


That answer gives the general method for Lagrange multipliers. We can 
apply it to the problem you quoted:

We have the total cost

     f(x,y,z) = 5xy + 4yz + 8zx

and the constraint is

     g(x,y,z) =  xyz - 2500 = 0

The auxiliary function is  

     phi(x,y,z) = f(x,y,z) - kg(x,y,z)

     part[d(phi)/dx] = 5y + 8z - k[yz] = 0  .....................[1]
     part[d(phi)/dy] = 5x + 4z - k[xz] = 0  .....................[2]
     part[d(phi)/dz] = 4y + 8x - k[xy] = 0  .....................[3]
     g(x,y,z) = 0           xyz - 2500 = 0  .....................[4]

and from these 4 equations we must find k, x, y, z. 

     Multiply [1] by x  -> 5xy + 8zx - k(xyz) = 0  ..............[5]
        "     [2]    y  -> 5xy + 4yz - k(xyz) = 0  ..............[6]
        "     [3]    z  -> 4yz + 8zx - k(xyz) = 0  ..............[7]

  [5] - [6]  ->    8zx-4yz = 0
                  4z(2x-y) = 0   and so   y = 2x  ...............[8]

  [5] - [7]  ->    5xy-4yz = 0   
                  y(5x-4z) = 0   and so   z = 5x/4  .............[9]

Substitute [8] and [9] into xyz = 2500

                    x(2x)(5x/4) = 2500

                      (10/4)x^3 = 2500

                            x^3 = 1000

     and so                   x = 10

Then y = 20 and z = 12.5

The best dimensions for the box are 10 x 20 x 12.5  cm^3.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/