Explanation Given: The rectangular package shall have the sum of its length and girth not exceed 120 inches. Calculation: Let x be a number of kilograms of type M food and y be a number of kilograms of type N food. Since the type M cost $1 per kilograms and type N costs $2 per kilograms, the weekly food costs is f ( x , y ) = x + 2 y . Since the group of guinea pigs is to receive 25,600 calories per week and two available foods produce 200 xy calories, the constraint is 200 x y = 25 , 600 . 200 x y − 25 , 600 = 0 Minimize cost f ( x , y ) = x + 2 y subject to the constraint g ( x , y ) = 200 x y − 25 , 600 . Need to form a function F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) . Substitute f ( x , y ) = x + 2 y and g ( x , y ) = 200 x y − 25 , 600 in F ( x , y , λ ) , F ( x , y , λ ) = x + 2 y + λ ( 200 x y − 25 , 600 ) = x + 2 y + 200 x y λ − 25 , 600 λ Thus, the required function is F ( x , y , λ ) = x + 2 y + 200 x y λ − 25 , 600 λ . The partial derivative of the function F ( x , y , λ ) with respect to x is computed below. F x ( x , y , λ ) = ∂ ∂ x ( x + 2 y + 200 x y λ − 25 , 600 λ ) = ∂ ∂ x ( x ) + ∂ ∂ x ( 2 y ) + ∂ ∂ x ( 200 x y λ ) − ∂ ∂ x ( 25 , 600 λ ) = 1 + 0 + ( 200 y λ ) ∂ ∂ x ( x ) − 0 = 1 + 200 y λ Therefore, the partial derivative of F with respect to x is F x ( x , y , λ ) = 1 + 200 y λ . The partial derivative of the function F ( x , y , λ ) with respect to y is computed below. F y ( x , y , λ ) = ∂ ∂ y ( x + 2 y + 200 x y λ − 25 , 600 λ ) = ∂ ∂ y ( x ) + ∂ ∂ y ( 2 y ) + ∂ ∂ y ( 200 x y λ ) − ∂ ∂ y ( 25 , 600 λ ) = 0 + 2 ( 1 ) + ( 200 x λ ) 1 ( 1 ) − 0 = 2 + 200 x λ Thus, the partial derivative of the function F with respect to y is F y ( x , y , λ ) = 2 + 200 x λ

Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

14th Edition

ISBN: 9780137553341

Author: Raymond Barnett, Michael Ziegler

Publisher: PEARSON+

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Chapter 14.4, Problem 38E

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To find: The values of x and y that minimize the weekly food cost as well as find the minimum cost.

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Chapter 14.4, Problem 37E

Chapter 14.5, Problem 1MP

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Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

Ch. 14.1 - Matched Problem 1 Find C(20, 10) for the cost...Ch. 14.1 - Prob. 2MP Ch. 14.1 - Prob. 3MP Ch. 14.1 - Prob. 4MP Ch. 14.1 - Prob. 5MP Ch. 14.1 - Prob. 6MP Ch. 14.1 - Prob. 7MP Ch. 14.1 - Prob. 1ED Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E

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