Explanation Given: Area of an open rectangular box is 48 ft 2 constructed from a piece of a cardboard. Formula used: Lagrange Multipliers: To obtain the relative extremum of the function f ( x , y ) subject to the constraint g ( x , y ) = 0 (assuming that these extreme values exist), (1) Form an auxiliary function F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) 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 and F λ = 0 for all values of x , y , 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. Surface area of an open rectangular box is x y + 2 ( y z + x z ) . The area of an open rectangular box is 48 ft 2 . x y + 2 ( y z + x z ) = 48 x y + 2 ( y z + x z ) − 48 = 0 The Volume of a rectangular box is x y z . Obtain the maximum of f ( x , y , z ) = x y z constrained to the function g ( x , y , z ) = x y + 2 ( y z + x z ) − 48 . 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 y + 2 ( y z + x z ) − 48 ) The above expression is called the Lagrange function while λ be the Lagrange multipliers. Differentiate F ( x , y , λ ) = x y z + λ ( x y + 2 ( y z + x z ) − 48 ) with respect to x , d d x ( F ( x , y , λ ) ) = d d x ( x y z + λ ( x y + 2 ( y z + x z ) − 48 ) ) F x = d d x ( x y z ) + d d x ( λ ( x y + 2 ( y z + x z ) − 48 ) ) = y z + y λ + 2 z λ Differentiate F ( x , y , λ ) = x y z + λ ( x y + 2 ( y z + x z ) − 48 ) with respect to y , d d y ( F ( x , y , λ ) ) = d d y ( x y z + λ ( x y + 2 ( y z + x z ) − 48 ) ) F y = d d y ( x y z ) + d d y ( λ ( x y + 2 ( y z + x z ) − 48 ) ) = x z + x λ + 2 z λ Differentiate F ( x , y , λ ) = x y z + λ ( x y + 2 ( y z + x z ) − 48 ) with respect to z , d d z ( F ( x , y , λ ) ) = d d z ( x y z + λ ( x y + 2 ( y z + x z ) − 48 ) ) F z = d d z ( x y z ) + d d z ( λ ( x y + 2 ( y z + x z ) − 48 ) ) = x y + 2 y λ + 2 x λ Differentiate F ( x , y , λ ) = x y z

Applied Calculus for the Managerial, Life, and Social Sciences

10th Edition

ISBN: 9781305887831

Author: Tan

Publisher: CENGAGE CO

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Chapter 8.5, Problem 20E

To determine

To find: The dimension a rectangular box of having maximum volume and the volume of the box.

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Chapter 8 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences

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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The dimension a rectangular box of having maximum volume and the volume of the box.

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Chapter 8 Solutions