Explanation Given: Two sides of a fence can be built for $6 per feet and other two side also built for $4. 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 area of the field is f ( x , y ) = x y where x is the length of the field and y is the width of the field. Since x is the number of feet in width of the rectangular field and y is also number of the field in length of field. Here, the maximum area can be enclosed for $1200. 6 x + 6 y + 4 x + 4 y = 1200 10 x + 10 y = 1200 Therefore, the constraint is 10 x + 10 y = 1200 . The function f ( x , y ) = x y is subject to the constraint 10 x + 10 y = 1200 . The constrain in the form g ( x , y ) = 0 . Thus, the given constrain can be written as g ( x , y ) = 10 x + 10 y − 1200 The form of the Lagrange function is, F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) . Substitute f ( x , y ) = x y and g ( x , y ) = 10 x + 10 y − 1200 in F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) , F ( x , y , λ ) = x y − λ ( 10 x + 10 y − 1200 ) = x y − 10 λ x − 10 λ y − 1200 λ Thus, the Lagrange function is F ( x , y , λ ) = x y − 10 λ x − 10 λ y − 1200 λ . Need to find the partial derivative of the function F ( x , y , λ ) . Partial derivative of the function F ( x , y , λ ) with respect to x and fix y and λ are constant. F x ( x , y , λ ) = ∂ ∂ x ( x y − 10 λ x − 10 λ y − 1200 λ ) = ∂ ∂ x ( x y ) − ∂ ∂ x ( 10 λ x ) − ∂ ∂ x ( 10 λ y ) + ∂ ∂ x ( 1200 λ ) = ( y ) ∂ ∂ x ( x ) − ( 10 λ ) ∂ ∂ x ( x ) − ∂ ∂ x ( 3 λ y ) + ∂ ∂ x ( 42 λ ) Simplify the terms and obtain the partial derivatives, F x ( x , y , λ ) = ( y ) ( 1 ) − ( 10 λ ) ( 1 ) − 0 + 0 = y − 10 λ Therefore, the partial derivative with respect to x is F x ( x , y , λ ) = y − 10 λ . Partial derivative of the function F ( x , y , λ ) with respect to y and fix x and λ are constant

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 27E

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Chapter 17.4, Problem 26E

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Chapter 17 Solutions

Finite Mathematics and Calculus with Applications (10th Edition)

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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 maximum area if the area enclosed for $ 1200.

Solution Summary: The author calculates the maximum area if the area enclosed for 1200. The form of the constraint is g(x,y)=0.

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Chapter 17 Solutions