Explanation Given: The combined length and girth cannot exceed 108 in for a rectangular package. Formula used: Lagrange Multipliers: To obtain the relative extremum of the function f ( x , y , z ) subject to the constraint g ( x , y , z ) = 0 (assuming that these extreme values exist), (1) Form an auxiliary function F ( x , y , z , λ ) = f ( x , y , z ) + λ g ( x , y , z ) called the Lagrangian function where the variable λ is called the Lagrange multiplier. (2) Solve the system that consists of the equations F x = 0 , F y = 0 , F z = 0 and F λ = 0 for all values of x , y , z and λ . (3) The solutions found in step (2) are candidates for the extremum of f . Calculation: Suppose the length, width and height of the rectangular box is x , y and z respectively. The combined length and girth of an open rectangular box is x + 2 ( y + z ) . The combined length and girth of an open rectangular box is 108. x + 2 ( y + z ) = 108 x + 2 ( y + z ) − 108 = 0 The Volume of a rectangular box is x y z . Suppose, g ( x , y , z ) = x + 2 ( y + z ) − 108 . Obtain the maximum of f ( x , y , z ) = x y z constrained to the function g ( x , y , z ) = x + 2 ( y + z ) − 108 . Hence, frame an auxiliary function F for the given function f ( x , y , z ) subject to g ( x , y , z ) = 0 . F ( x , y , λ ) = f ( x , y , z ) + g ( x , y , z ) = x y z + λ ( x + 2 ( y + z ) − 108 ) The above expression is called the Lagrange function while λ be the Lagrange multipliers. Differentiate F ( x , y , λ ) = x y z + λ ( x + 2 ( y + z ) − 108 ) with respect to x , d d x ( F ( x , y , λ ) ) = d d x ( x y z + λ ( x + 2 ( y + z ) − 108 ) ) F x = d d x ( x y z ) + d d x ( λ ( x + 2 ( y + z ) − 108 ) ) = y z + λ Differentiate F ( x , y , λ ) = x y z + λ ( x + 2 ( y + z ) − 108 ) with respect to y , d d y ( F ( x , y , λ ) ) = d d y ( x y z + λ ( x + 2 ( y + z ) − 108 ) ) F y = d d y ( x y z ) + d d y ( λ ( x + 2 ( y + z ) − 108 ) ) = x z + 2 λ Differentiate F ( x , y , λ ) = x y z + λ ( x + 2 ( y + z ) − 108 ) with respect to z , <

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 8.5, Problem 23E

To determine

To find: The dimension of a rectangular package having the greatest possible volume.

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Chapter 8 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

Ch. 8.1 - What is a function of two variables? Give an...Ch. 8.1 - Prob. 2CQ Ch. 8.1 - Prob. 3CQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Let f(x, y) = x2+ 2xy x + 3. Compute f(1, 2),...Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Let h(s, t) = s ln t t ln s. Compute h(1, e),...

Ch. 8.1 - Prob. 8E Ch. 8.1 - Let g(r, s, t) = res/t. Compute g(1, 1, 1), g(1,...Ch. 8.1 - Prob. 10E Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - In Exercises 11-18, find the domain of the...Ch. 8.1 - Prob. 14E Ch. 8.1 - In Exercises 11-18, find the domain of the...Ch. 8.1 - Prob. 16E Ch. 8.1 - In Exercises 11-18, find the domain of the...Ch. 8.1 - Prob. 18E Ch. 8.1 - Prob. 19E Ch. 8.1 - Prob. 20E Ch. 8.1 - In Exercises 19-24, sketch the level curves of the...Ch. 8.1 - Prob. 22E Ch. 8.1 - In Exercises 1924, sketch the level curves of the...Ch. 8.1 - Prob. 24E Ch. 8.1 - Find an equation of the level curve of f (x, y) =...Ch. 8.1 - Prob. 26E Ch. 8.1 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Prob. 30E Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - Prob. 33E Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - POISEITLLES LAW Poiseuilles Law states that the...Ch. 8.1 - COST FUNCTION FOR A LOUDSPEAKER SYSTEM Acrosonic...Ch. 8.1 - Prob. 38E Ch. 8.1 - REVENUE FUNCTIONS FOR DESKS Country Workshop...Ch. 8.1 - Prob. 40E Ch. 8.1 - 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Prob. 30E Ch. 8.5 - Prob. 31E Ch. 8.5 - Prob. 33E Ch. 8.5 - Prob. 34E Ch. 8.5 - Prob. 36E Ch. 8.5 - Prob. 38E Ch. 8.5 - Prob. 39E Ch. 8.5 - Prob. 40E Ch. 8.5 - Prob. 41E Ch. 8.5 - Prob. 42E Ch. 8.6 - Prob. 1CQ Ch. 8.6 - Prob. 2CQ Ch. 8.6 - Prob. 1E Ch. 8.6 - Prob. 2E Ch. 8.6 - Prob. 3E Ch. 8.6 - Prob. 4E Ch. 8.6 - Prob. 5E Ch. 8.6 - Prob. 6E Ch. 8.6 - Prob. 7E Ch. 8.6 - Prob. 8E Ch. 8.6 - Prob. 9E Ch. 8.6 - Prob. 10E Ch. 8.6 - Prob. 11E Ch. 8.6 - Prob. 12E Ch. 8.6 - Prob. 13E Ch. 8.6 - Prob. 14E Ch. 8.6 - In Exercises 318, find the total differential of...Ch. 8.6 - Prob. 16E Ch. 8.6 - Prob. 17E Ch. 8.6 - Prob. 18E Ch. 8.6 - Prob. 19E Ch. 8.6 - Prob. 20E Ch. 8.6 - Prob. 21E Ch. 8.6 - Prob. 22E Ch. 8.6 - Prob. 23E Ch. 8.6 - Prob. 24E Ch. 8.6 - Prob. 25E Ch. 8.6 - Prob. 26E Ch. 8.6 - Prob. 27E Ch. 8.6 - Prob. 28E Ch. 8.6 - Prob. 29E Ch. 8.6 - Prob. 30E Ch. 8.6 - Prob. 31E Ch. 8.6 - Prob. 32E Ch. 8.6 - Prob. 33E Ch. 8.6 - Prob. 34E Ch. 8.6 - Prob. 35E Ch. 8.6 - Prob. 36E Ch. 8.6 - Prob. 37E Ch. 8.6 - 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The dimension of a rectangular package having the greatest possible volume.

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To find: The dimension of a rectangular package having the greatest possible volume.

Chapter 8 Solutions