Explanation Given information: a box has of its three of faces in the coordinate planes and one vertex on the plane 2 x + 3 y + 4 z = 6 . Calculation: From the given information, it is a typical optimization problem with constraint, where the vertex of the box is constrained on the plane 2 x + 3 y + 4 z = 6 and they need to find the maximize the volume ( V ) of the box. So, will proceed in this problem using the Lagrange multiplier method is given below: 2 x + 3 y + 4 z = 6................................ ( 1 ) Now invoking the Lagrange multiplier method ∴ F = x y z + λ ( 2 x + 3 y + 4 z ) Taking the partial derivative w.r.t x , y , and z ∴ ∂ F ∂ x = y z + 2 λ = 0....................... ( 2 ) ∴ ∂ F ∂ y = x z + 3 λ = 0........................ ( 3 ) ∴ ∂ F ∂ z = x y + 4 λ = 0......................... ( 4 ) Now with three partial derivative equations and the original constraints equation, will be solve for unknown proportions

Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

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Question

Chapter 4.9, Problem 6P

To determine

To calculate: The maximum value of the box.

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Chapter 4.9, Problem 5P

Chapter 4.9, Problem 7P

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Chapter 4 Solutions

Mathematical Methods in the Physical Sciences

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