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/ |
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