Explanation Given: The given function is P ( x , y ) = 0.01 ( − x 2 + 3 x y + 160 x − 5 y 2 + 200 y + 2600 ) with a budget constraint of x + y = 280 . Calculation: The Lagrange’s Method is used to optimize the function f ( x , y ) subject to the constraint g ( x , y ) = 0 . Here f ( x , y ) = P ( x , y ) and g ( x , y ) is x + y − 280 = 0 . Form the Lagrange function: F ( x, y, λ ) =f ( x, y ) -λg ( x, y ) Replace as per the given values: F(x, y, λ)=0 .01(-x 2 +3xy+160x-5y 2 +200y+2600)-λ(x+y-280) =0 .01(-x 2 +3xy+160x-5y 2 +200y+2600)-λx-λy+280λ Power rule states that f ( x ) = x n for any number n , then f ′ ( x ) = n x n − 1 . Use the power rule and find the partial derivatives of F as: F x ( x , y , λ ) , F y ( x , y , λ ) , F λ ( x , y , λ ) Again, replace the values and solve: F x (x, y, λ)=0 .01(-2x+3y+160)-λ =0 .01(-2x+3y)+1 .6-λ F y (x, y, λ)=0 .01(3x-10y+200)-λ =0 .01(3x-10y)+2-λ And, F λ ( x , y , λ ) = − ( x + y − 280 ) Solve the below mentioned system of equations: F x ( x , y , λ ) = 0 F y ( x , y , λ ) = 0 F λ ( x , y , λ ) = 0 Set each partial derivative equal to 0 and solve the resultant system: 0.01 ( − 2 x + 3 y ) + 1.6 − λ = 0 0.01 ( 3 x − 10 y ) + 2 − λ = 0 − ( x + y − 280 ) = 0 Solve the first two equations for λ : λ = 0.01 ( − 2 x + 3 y ) + 1.6 λ = 0.01 ( 3 x − 10 y ) + 2 Set the two expressions for λ equal to obtain the equation: 0

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

12th Edition

ISBN: 9780134767628

Author: Margaret L. Lial, Thomas W. Hungerford, John P. Holcomb, Bernadette Mullins

Publisher: PEARSON

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Chapter 14, Problem 54RE

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To calculate: The amount spent on fertilizer and seed that will maximize profit by Lagrange’s method using function P(x,y)=0.01(−x2+3xy+160x−5y2+200y+2600).

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Chapter 14, Problem 53RE

Chapter 14, Problem 1CE

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Chapter 14 Solutions

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

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Checkout Lines According to...Ch. 14.1 - Prob. 40E Ch. 14.1 - Prob. 41E Ch. 14.1 - Prob. 42E Ch. 14.1 - Prob. 43E Ch. 14.1 - Prob. 44E Ch. 14.1 - Prob. 45E Ch. 14.1 - Prob. 46E Ch. 14.1 - Prob. 47E Ch. 14.1 - Prob. 48E Ch. 14.2 - Prob. 1CP Ch. 14.2 - Prob. 2CP Ch. 14.2 - Prob. 3CP Ch. 14.2 - Prob. 4CP Ch. 14.2 - Prob. 5CP Ch. 14.2 - Prob. 6CP Ch. 14.2 - Prob. 7CP Ch. 14.2 - Prob. 8CP Ch. 14.2 - Prob. 9CP Ch. 14.2 - Prob. 1E Ch. 14.2 - For each of the given functions, find the...Ch. 14.2 - Prob. 3E Ch. 14.2 - Prob. 4E Ch. 14.2 - Prob. 5E Ch. 14.2 - Prob. 6E Ch. 14.2 - Prob. 7E Ch. 14.2 - Prob. 8E Ch. 14.2 - Prob. 9E Ch. 14.2 - Prob. 10E Ch. 14.2 - Prob. 11E Ch. 14.2 - Prob. 12E Ch. 14.2 - Prob. 13E Ch. 14.2 - Prob. 14E Ch. 14.2 - Prob. 15E Ch. 14.2 - Prob. 16E Ch. 14.2 - Prob. 17E Ch. 14.2 - Prob. 18E Ch. 14.2 - Prob. 19E Ch. 14.2 - Prob. 20E Ch. 14.2 - Prob. 21E Ch. 14.2 - Prob. 22E Ch. 14.2 - Prob. 23E Ch. 14.2 - Prob. 24E Ch. 14.2 - Prob. 25E Ch. 14.2 - Prob. 26E Ch. 14.2 - Prob. 27E Ch. 14.2 - Prob. 28E Ch. 14.2 - Prob. 29E Ch. 14.2 - Prob. 30E Ch. 14.2 - Prob. 31E Ch. 14.2 - Prob. 32E Ch. 14.2 - Prob. 33E Ch. 14.2 - Prob. 34E Ch. 14.2 - Prob. 35E Ch. 14.2 - Prob. 36E Ch. 14.2 - Prob. 37E Ch. 14.2 - Find fx,fy,fz, and fyz for the given functions. In...Ch. 14.2 - Prob. 39E Ch. 14.2 - Suppose z=f(x,y) describes the cost to build a...Ch. 14.2 - Prob. 41E Ch. 14.2 - 42. Business The revenue from the sale of x units...Ch. 14.2 - Prob. 62E Ch. 14.2 - Prob. 44E Ch. 14.2 - Prob. 45E Ch. 14.2 - Prob. 43E Ch. 14.2 - Prob. 42E Ch. 14.2 - Prob. 59E Ch. 14.2 - Prob. 50E Ch. 14.2 - Prob. 51E Ch. 14.2 - Work the following applied problems. (See Example...Ch. 14.2 - Prob. 53E Ch. 14.2 - Work the following marginal productivity problems....Ch. 14.2 - Prob. 55E Ch. 14.2 - Prob. 56E Ch. 14.2 - Prob. 58E Ch. 14.2 - Work the following applied problems. 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Find positive numbers x and y whose sum is 80...Ch. 14 - Prob. 52RE Ch. 14 - Prob. 53RE Ch. 14 - Prob. 54RE Ch. 14 - Prob. 1CE Ch. 14 - Prob. 2CE

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The amount spent on fertilizer and seed that will maximize profit by Lagrange’s method using function P ( x , y ) = 0.01 ( − x 2 + 3 x y + 160 x − 5 y 2 + 200 y + 2600 ) .

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To calculate: The amount spent on fertilizer and seed that will maximize profit by Lagrange’s method using function P(x,y)=0.01(−x2+3xy+160x−5y2+200y+2600).

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Chapter 14 Solutions