Explanation Definition used: “The Lagrange multipliers defined as ∇ f ( x , y , z ) = λ ∇ g ( x , y , z ) . This equation can be expressed as f x = λ g x , f y = λ g y , f z = λ g z and g ( x , y , z ) = k ”. Calculation: The surface area is f ( x , y , z ) = 2 x y + 2 y z + 2 x z , where x, y and z are dimensions. The volume of the rectangular box is V = x y z = 16 . Then, the value of the function f ( x , y , z ) = 2 x y + 2 y z + 2 x z subject to the constraint g ( x , y , z ) = x y z = 16 is used in Lagrange multipliers. Compute the Lagrange multipliers ∇ f ( x , y ) = λ ∇ g ( x , y ) . ∇ f ( x , y , z ) = λ ∇ g ( x , y , z ) 〈 f x , f y , f z 〉 = λ 〈 g x , g y , g z 〉 〈 f x ( 2 x y + 2 y z + 2 x z ) , f y ( 2 x y + 2 y z + 2 x z ) , f z ( 2 x y + 2 y z + 2 x z ) 〉 = λ 〈 g x ( x y z ) , g y ( x y z ) , g z ( x y z ) 〉 〈 2 y + 2 z , 2 x + 2 z , 2 y + 2 x 〉 = λ 〈 y z , x z , x y 〉 Thus, the value of ∇ f ( x , y , z ) = λ ∇ g ( x , y , z ) is 〈 2 y + 2 z , 2 x + 2 z , 2 y + 2 x 〉 = λ 〈 y z , x z , x y 〉 . By the definition, 〈 2 y + 2 z , 2 x + 2 z , 2 y + 2 x 〉 = λ 〈 y z , x z , x y 〉 is expressed as follows. 2 y + 2 z = λ ( y z ) (1) 2 x + 2 z = λ ( x z ) (2) 2 y + 2 x = λ ( x y ) (3) x y z = 16 (4) From the equations (1), obtain the following. 2 y + 2 z = λ ( y z ) λ = 2 ( y + z ) y z λ = 2 z + 2 y From the equations (2), obtain the following. 2 x + 2 z = λ ( x z ) λ = 2 ( x + z ) x z λ = 2 z + 2 x From the equations (3), obtain the following

2nd Edition

ISBN: 9780136564133

Author: Pearson

Publisher: PEARSON C

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Question

Chapter 15.8, Problem 28E

To determine

To find: The rectangular box of volume 16 ft3 which has minimum surface area with the help of Lagrange multipliers.

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Chapter 15.8, Problem 27E

Chapter 15.8, Problem 29E

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A rectangular storage tank with an open top is to have volume (20 m³). The length of the base is twice the width, see the figure. The Material of the base costs (20 $) per square meter. The material of the sides costs (10 $) per square meter. Use Lagrange multiplier method to find the optimum dimensions of a tank that give the minimum cost. h 2w Ans./ l= 3.914 W = 1.957 h - 2.61

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Your company are designing metal trays. They will be made out of a sheet of metalwhich has a perimeter of 80 cm. The box is made by removing squares of side y = 2cm from each corner of the rectangle and then folding in the sides. A net is shownbelow.● Your company wants to maximise the capacity of these trays and wants you toinvestigate what value of x will give you the maximum volume.● Investigate how changing the size of the square at each corner would affect thecapacity of the tray and investigate the value of y which would give youmaximum volume.