Explanation Given: Open on top with a volume of 32 ft 3 . Definition used: Lagrange Multipliers 1. The form of the constraint is g ( x , y ) = 0 . 2. The form of the Lagrange function F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) . 3 Form the system of equations F x ( x , y , λ ) = 0 , F y ( x , y , λ ) = 0 and F λ ( x , y , λ ) = 0 , compute the solution of these system also the relative extrema among these solution. Calculation: The rectangular fish aquarium is open on top. Let the dimension of the bottom is x by y . The height of the rectangular aquarium is z . The volume of the b fish aquarium is V = x 2 z . Since the open on top with a volume of 32 ft 3 . 32 = x 2 z x 2 z − 32 = 0 The surface area of the fish aquarium is, f ( x , y ) = x y + 2 x z + 2 y z . Thus, the multivariable function f ( x , y ) = x y + 2 x z + 2 y z . The constrain in the form g ( x , y , z ) = 0 . Thus, the given constrain can be written as g ( x , y , z ) = x 2 z − 32 . The surface area of the box is, f ( x , y ) = x y + 2 x z + 2 y z . The function f ( x , y ) = x y + 2 x z + 2 y z subject to the constraint g ( x , y , z ) = x 2 z − 32 . The form of the Lagrange function is, F ( x , y , z , λ ) = f ( x , y , z ) − λ ⋅ g ( x , y , z ) . Substitute f ( x , y ) = x y + 2 x z + 2 y z and g ( x , y , z ) = x 2 z − 32 in F ( x , y , z , λ ) = f ( x , y , z ) − λ ⋅ g ( x , y , z ) , F ( x , y , z , λ ) = x y + 2 x z + 2 y z − λ ( x 2 z − 32 ) = x y + 2 x z + 2 y z − x 2 z λ + 32 λ Thus, the Lagrange function is F ( x , y , z , λ ) = x y + 2 x z + 2 y z − x 2 z λ + 32 λ . Here need to find the partial derivative of the function F ( x , y , z , λ ) . Partial derivative of the function F ( x , y , z , λ ) with respect to x and fix y , z and λ are constant. F x ( x , y , z , λ ) = ∂ ∂ x ( x y + 2 x z + 2 y z − x y z λ + 32 λ ) = ∂ ∂ x ( x y ) + ∂ ∂ x ( 2 x z ) + ∂ ∂ x ( 2 y z ) − ∂ ∂ x ( x y z λ ) + ∂ ∂ x ( 32 λ ) = ( y ) ∂ ∂ x ( x ) + ( 2 z ) ∂ ∂ x ( x ) + ∂ ∂ x ( 2 y z ) − ( z λ y ) ∂ ∂ x ( x ) + ∂ ∂ x ( 32 λ ) Simplify the terms and obtain the partial derivatives, F x ( x , y , z , λ ) = ( y ) ( 1 ) + ( 2 z ) ( 1 ) + 0 − ( z λ y ) ( 1 ) + 0 = y + 2 z − z λ y Therefore, the partial derivative with respect to x is F x ( x , y , z , λ ) = y + 2 z − z λ y . Partial derivative of the function F ( x , y , z , λ ) with respect to y and fix x , z and λ are constant. F y ( x , y , z , λ ) = ∂ ∂ y ( x y + 2 x z + 2 y z − x y z λ + 32 λ ) = ∂ ∂ y ( x y ) + ∂ ∂ y ( 2 x z ) + ∂ ∂ y ( 2 y z ) − ∂ ∂ y ( x y z λ ) + ∂ ∂ y ( 32 λ ) = ( x ) ∂ ∂ y ( y ) + ∂ ∂ y ( 2 x z ) + ( 2 z ) ∂ ∂ y ( y ) − ∂ ∂ y ( x y z λ ) + 0 Simplify the terms and obtain the partial derivatives, F y ( x , y , z , λ ) = ( x ) ∂ ∂ y ( y ) + ∂ ∂ y ( 2 x z ) + ( 2 z ) ∂ ∂ y ( y ) − ∂ ∂ y ( x y z λ ) + 0 = ( x ) ( 1 ) + 0 + ( 2 z ) ( 1 ) − ( x z λ ) ∂ ∂ y ( y ) + 0 = x + 2 z − x z λ Therefore, the partial derivative with respect to y is F y ( x , y , z , λ ) = x + 2 z − x z λ . Partial derivative of the function F ( x , y , z , λ ) with respect to z and fix x , y and λ are constant. F z ( x , y , z , λ ) = ∂ ∂ z ( x y + 2 x z + 2 y z − x y z λ + 32 λ ) = ∂ ∂ z ( x y ) + ∂ ∂ z ( 2 x z ) + ∂ ∂ z ( 2 y z ) − ∂ ∂ z ( x y z λ ) + ∂ ∂ z ( 32 λ ) = ∂ ∂ z ( x y ) + ( 2 x ) ∂ ∂ z ( z ) + ( 2 y ) ∂ ∂ z ( z ) − ( x y λ ) ∂ ∂ z ( z ) + ∂ ∂ z ( 32 λ ) Simplify the terms and obtain the partial derivatives, F z ( x , y , z , λ ) = 0 + ( 2 x ) ( 1 ) + ( 2 y ) ( 1 ) − ( x y λ ) ( 1 ) + 0 = 2 x − 2 y − x y λ Therefore, the partial derivative with respect to y is F z ( x , y , z , λ ) = 2 x − 2 y − x y λ

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 17.4, Problem 40E

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Chapter 17.4, Problem 39E

Chapter 17.4, Problem 41E

Chapter 17 Solutions

Finite Mathematics and Calculus with Applications (10th Edition)

Ch. 17.1 - For the function in Example 1, find f(2, 3)....Ch. 17.1 - Prob. 2YT Ch. 17.1 - Prob. 3YT Ch. 17.1 - Prob. 4YT Ch. 17.1 - Prob. 1WE Ch. 17.1 - Prob. 2WE Ch. 17.1 - Prob. 3WE Ch. 17.1 - Prob. 4WE Ch. 17.1 - Prob. 5WE Ch. 17.1 - Prob. 6WE

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W2.23x+6dx Ch. 17.6 - Prob. 3WE Ch. 17.6 - Prob. 4WE Ch. 17.6 - Prob. 5WE Ch. 17.6 - Prob. 6WE Ch. 17.6 - Prob. 1E Ch. 17.6 - Prob. 2E Ch. 17.6 - Prob. 3E Ch. 17.6 - Prob. 4E Ch. 17.6 - Prob. 5E Ch. 17.6 - Prob. 6E Ch. 17.6 - Prob. 7E Ch. 17.6 - Prob. 8E Ch. 17.6 - Prob. 9E Ch. 17.6 - Prob. 10E Ch. 17.6 - Prob. 11E Ch. 17.6 - Prob. 12E Ch. 17.6 - Prob. 13E Ch. 17.6 - Prob. 14E Ch. 17.6 - Prob. 15E Ch. 17.6 - Prob. 16E Ch. 17.6 - Prob. 17E Ch. 17.6 - Prob. 18E Ch. 17.6 - Prob. 19E Ch. 17.6 - Prob. 20E Ch. 17.6 - Prob. 21E Ch. 17.6 - Prob. 22E Ch. 17.6 - Prob. 23E Ch. 17.6 - Prob. 24E Ch. 17.6 - Prob. 25E Ch. 17.6 - Prob. 26E Ch. 17.6 - Prob. 27E Ch. 17.6 - Prob. 28E Ch. 17.6 - Prob. 29E Ch. 17.6 - Prob. 30E Ch. 17.6 - Prob. 31E Ch. 17.6 - Prob. 32E Ch. 17.6 - Prob. 33E Ch. 17.6 - Prob. 34E Ch. 17.6 - Prob. 35E Ch. 17.6 - Prob. 36E Ch. 17.6 - Prob. 37E Ch. 17.6 - Prob. 38E Ch. 17.6 - Prob. 39E Ch. 17.6 - Prob. 40E Ch. 17.6 - Prob. 41E Ch. 17.6 - Prob. 42E Ch. 17.6 - Prob. 43E Ch. 17.6 - Use the region R with the indicated boundaries to...Ch. 17.6 - Prob. 45E Ch. 17.6 - Prob. 46E Ch. 17.6 - Prob. 47E Ch. 17.6 - Prob. 48E Ch. 17.6 - Prob. 49E Ch. 17.6 - Prob. 50E Ch. 17.6 - Prob. 51E Ch. 17.6 - Prob. 52E Ch. 17.6 - Prob. 53E Ch. 17.6 - Prob. 54E Ch. 17.6 - Prob. 55E Ch. 17.6 - Prob. 56E Ch. 17.6 - Prob. 57E Ch. 17.6 - Prob. 58E Ch. 17.6 - Prob. 59E Ch. 17.6 - Prob. 60E Ch. 17.6 - Prob. 61E Ch. 17.6 - Prob. 62E Ch. 17.6 - Prob. 63E Ch. 17.6 - Prob. 64E Ch. 17.6 - Prob. 65E Ch. 17.6 - Prob. 66E Ch. 17.6 - Prob. 67E Ch. 17.6 - Prob. 68E Ch. 17.6 - Prob. 69E Ch. 17.6 - Time In Exercise 39 of Section 17.3, we saw that...Ch. 17.6 - Profit In Exercise 38 of Section 17.3, we saw that...Ch. 17 - Determine whether each of the following statements...Ch. 17 - Prob. 2RE Ch. 17 - Prob. 3RE Ch. 17 - Prob. 4RE Ch. 17 - Prob. 5RE Ch. 17 - Prob. 6RE Ch. 17 - Prob. 7RE Ch. 17 - Prob. 8RE Ch. 17 - Prob. 9RE Ch. 17 - Prob. 10RE Ch. 17 - Prob. 11RE Ch. 17 - Prob. 12RE Ch. 17 - Prob. 13RE Ch. 17 - Prob. 14RE Ch. 17 - Prob. 15RE Ch. 17 - Prob. 16RE Ch. 17 - Prob. 17RE Ch. 17 - Prob. 18RE Ch. 17 - Prob. 19RE Ch. 17 - Prob. 20RE Ch. 17 - Prob. 21RE Ch. 17 - Prob. 22RE Ch. 17 - Prob. 23RE Ch. 17 - Prob. 24RE Ch. 17 - Prob. 25RE Ch. 17 - Prob. 26RE Ch. 17 - Prob. 27RE Ch. 17 - Prob. 28RE Ch. 17 - Prob. 29RE Ch. 17 - Prob. 30RE Ch. 17 - Prob. 31RE Ch. 17 - Prob. 32RE Ch. 17 - Prob. 33RE Ch. 17 - Prob. 34RE Ch. 17 - Prob. 35RE Ch. 17 - Prob. 36RE Ch. 17 - Prob. 37RE Ch. 17 - Prob. 38RE Ch. 17 - Prob. 39RE Ch. 17 - Prob. 40RE Ch. 17 - Prob. 41RE Ch. 17 - Prob. 42RE Ch. 17 - Prob. 43RE Ch. 17 - Prob. 44RE Ch. 17 - Prob. 45RE Ch. 17 - Prob. 46RE Ch. 17 - Prob. 47RE Ch. 17 - Prob. 48RE Ch. 17 - Prob. 49RE Ch. 17 - Prob. 50RE Ch. 17 - Prob. 51RE Ch. 17 - Prob. 52RE Ch. 17 - Prob. 53RE Ch. 17 - Prob. 54RE Ch. 17 - Prob. 55RE Ch. 17 - Prob. 56RE Ch. 17 - Prob. 57RE Ch. 17 - Prob. 58RE Ch. 17 - Evaluate dz using the given information. 59.z =...Ch. 17 - Prob. 60RE Ch. 17 - Prob. 61RE Ch. 17 - Prob. 62RE Ch. 17 - Prob. 63RE Ch. 17 - Prob. 64RE Ch. 17 - Prob. 65RE Ch. 17 - Prob. 66RE Ch. 17 - Prob. 67RE Ch. 17 - Prob. 68RE Ch. 17 - Prob. 69RE Ch. 17 - Prob. 70RE Ch. 17 - Prob. 71RE Ch. 17 - Prob. 72RE Ch. 17 - Prob. 73RE Ch. 17 - Prob. 74RE Ch. 17 - Prob. 75RE Ch. 17 - Prob. 76RE Ch. 17 - Prob. 77RE Ch. 17 - Prob. 78RE Ch. 17 - Prob. 79RE Ch. 17 - Prob. 80RE Ch. 17 - Prob. 81RE Ch. 17 - Prob. 82RE Ch. 17 - Prob. 83RE Ch. 17 - Evaluate each double integral. If the function...Ch. 17 - Prob. 85RE Ch. 17 - Prob. 86RE Ch. 17 - Prob. 87RE Ch. 17 - Prob. 88RE Ch. 17 - Prob. 89RE Ch. 17 - Prob. 90RE Ch. 17 - Prob. 91RE Ch. 17 - Prob. 92RE Ch. 17 - Prob. 93RE Ch. 17 - Prob. 94RE Ch. 17 - Prob. 95RE Ch. 17 - Prob. 96RE Ch. 17 - Profit The total profit from 1 acre of a certain...Ch. 17 - Prob. 98RE Ch. 17 - Prob. 99RE Ch. 17 - Prob. 100RE Ch. 17 - Prob. 101RE Ch. 17 - Prob. 102RE Ch. 17 - Prob. 103RE Ch. 17 - Prob. 104RE Ch. 17 - Prob. 105RE Ch. 17 - Prob. 106RE

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The dimensions of rectangular fish aquarium closed box if it is minimize the surface area.

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Chapter 17 Solutions