Explanation Result used: Let the partial derivatives f x x , f y y and f x y of the function z = f ( x , y ) exists in the circular region contained in the xy -plane with center ( a , b ) and f x ( a , b ) = 0 , f y ( a , b ) = 0 and the discriminant D = f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 . (a) The function has relative maximum at ( a , b ) if D > 0 and f x x ( a , b ) < 0 . (b) The function has relative minimum at ( a , b ) if D > 0 and f x x ( a , b ) > 0 . (c) If D < 0 , then f ( a , b ) is a saddle point (neither a maximum nor a minimum) (d) If D = 0 , then the test fail. Calculation: The given revenue function is, P ( x , y ) = 36 x y − x 3 − 8 y 3 . Calculate the partial derivative of the above function with respect to x . P x ( x , y ) = ∂ ∂ x ( 36 x y − x 3 − 8 y 3 ) = ∂ ∂ x ( 36 x y ) − ∂ ∂ x ( x 3 ) − ∂ ∂ x ( 8 y 3 ) = 36 y − 3 x 2 − 0 = 36 y − 3 x 2 Again, take the partial derivative of the function P x ( x , y ) with respect to x . P x x ( x , y ) = ∂ ∂ x ( P x ( x , y ) ) = ∂ ∂ x ( 36 y − 3 x 2 ) = ∂ ∂ x ( 36 y ) − ∂ ∂ x ( 3 x 2 ) = − 6 x That is, P x x ( x , y ) = − 6 x (1) Calculate the partial derivative of the function P ( x , y ) with respect to y . P y ( x , y ) = ∂ ∂ y ( 36 x y − x 3 − 8 y 3 ) = ∂ ∂ y ( 36 x y ) − ∂ ∂ y ( x 3 ) − ∂ ∂ y ( 8 y 3 ) = 36 x − 0 − 24 y 2 = 36 x − 24 y 2 Again, take the partial derivative of the function P y ( x , y ) with respect to y . P y y ( x , y ) = ∂ ∂ y ( P y ( x , y ) ) = ∂ ∂ y ( 36 x − 24 y 2 ) = ∂ ∂ y ( 36 x ) − ∂ ∂ y ( 24 y 2 ) = − 48 y That is, P y y ( x , y ) = − 48 y (2) Calculate the partial derivative of the function P x ( x , y ) with respect to y . P x y ( x , y ) = ∂ ∂ y ( P x ( x , y ) ) = ∂ ∂ y ( 36 y − 3 x 2 ) = ∂ ∂ y ( 36 y ) − ∂ ∂ y ( 3 x 2 ) = 36 That is, P x y ( x , y ) = 36 (3) Obtain the critical point by equate the functions P x ( x , y ) = 0 and P y ( x , y ) = 0 . 36 y − 3 x 2 = 0 (4) 36 x − 24 y 2 = 0 (5) From equation (4), find the value of y in terms of x . 36 y − 3 x 2 = 0 36 y = 3 x 2 y = 3 36 x 2 y = 1 12 x 2 Substitute the value y = 1 12 x 2 in the equation (5)

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter 17.3, Problem 38E

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To find: The amount of steel x tons and aluminum y tons such that to maximize the profit function P(x,y)=36xy−x3−8y3 and use it to find the maximum profit.

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Chapter 17.3, Problem 37E

Chapter 17.3, Problem 39E

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Chapter 17 Solutions

Finite Mathematics and Calculus with Applications

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

Ch. 17.1 - Let f(x, y) = 2x 3y + 5. Find the following....Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E Ch. 17.1 - Prob. 7E Ch. 17.1 - Prob. 8E Ch. 17.1 - Graph the first-octant portion of each plane. 9.2x...Ch. 17.1 - Prob. 10E Ch. 17.1 - Prob. 11E Ch. 17.1 - Prob. 12E Ch. 17.1 - Prob. 13E Ch. 17.1 - Prob. 14E Ch. 17.1 - Prob. 15E Ch. 17.1 - Prob. 16E Ch. 17.1 - Prob. 17E Ch. 17.1 - Prob. 18E Ch. 17.1 - Prob. 19E Ch. 17.1 - Prob. 20E Ch. 17.1 - Prob. 21E Ch. 17.1 - Prob. 22E Ch. 17.1 - Match each equation in Exercises 2328 with its...Ch. 17.1 - Prob. 24E Ch. 17.1 - Prob. 25E Ch. 17.1 - Match each equation in Exercises 2328 with its...Ch. 17.1 - Prob. 27E Ch. 17.1 - Prob. 28E Ch. 17.1 - Prob. 29E Ch. 17.1 - Prob. 30E Ch. 17.1 - Prob. 31E Ch. 17.1 - Prob. 32E Ch. 17.1 - Prob. 33E Ch. 17.1 - Prob. 34E Ch. 17.1 - Production Find the level curve at a production of...Ch. 17.1 - Prob. 36E Ch. 17.1 - Prob. 37E Ch. 17.1 - Individual Retirement Accounts The multiplier...Ch. 17.1 - Individual Retirement Accounts The multiplier...Ch. 17.1 - Heat Loss The rate of heal loss (in watts) in...Ch. 17.1 - Prob. 41E Ch. 17.1 - Prob. 42E Ch. 17.1 - Pollution Intolerance According to research at the...Ch. 17.1 - Dengue Fever In tropical regions, dengue fever is...Ch. 17.1 - Prob. 45E Ch. 17.1 - Prob. 46E Ch. 17.1 - Prob. 47E Ch. 17.1 - Prob. 48E Ch. 17.1 - Prob. 49E Ch. 17.1 - Prob. 50E Ch. 17.1 - Prob. 51E Ch. 17.2 - Prob. 1YT Ch. 17.2 - Prob. 2YT Ch. 17.2 - Prob. 3YT Ch. 17.2 - Prob. 4YT Ch. 17.2 - Prob. 1WE Ch. 17.2 - Prob. 2WE Ch. 17.2 - Prob. 3WE Ch. 17.2 - Prob. 4WE Ch. 17.2 - Prob. 5WE Ch. 17.2 - Prob. 6WE Ch. 17.2 - Prob. 7WE Ch. 17.2 - Prob. 8WE Ch. 17.2 - Prob. 1E Ch. 17.2 - Prob. 2E Ch. 17.2 - Prob. 3E Ch. 17.2 - Prob. 4E Ch. 17.2 - Prob. 5E Ch. 17.2 - Prob. 6E Ch. 17.2 - Prob. 7E Ch. 17.2 - Prob. 8E Ch. 17.2 - Prob. 9E Ch. 17.2 - Prob. 10E Ch. 17.2 - Prob. 11E Ch. 17.2 - Prob. 12E Ch. 17.2 - Prob. 13E Ch. 17.2 - Prob. 14E Ch. 17.2 - Prob. 15E Ch. 17.2 - Prob. 16E Ch. 17.2 - 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Prob. 61E Ch. 17.2 - Prob. 62E Ch. 17.2 - Drug Reaction The reaction to x units of a drug t...Ch. 17.2 - Prob. 64E Ch. 17.2 - Prob. 66E Ch. 17.2 - Prob. 67E Ch. 17.2 - Prob. 68E Ch. 17.2 - Gravitational Attraction The gravitational...Ch. 17.2 - Velocity In 1931, Albert Einstein developed the...Ch. 17.2 - Prob. 71E Ch. 17.3 - Prob. 1YT Ch. 17.3 - Prob. 2YT Ch. 17.3 - Prob. 1WE Ch. 17.3 - Prob. 2WE Ch. 17.3 - Prob. 1E Ch. 17.3 - Prob. 2E Ch. 17.3 - Prob. 3E Ch. 17.3 - Prob. 4E Ch. 17.3 - Prob. 5E Ch. 17.3 - Prob. 6E Ch. 17.3 - Prob. 7E Ch. 17.3 - Prob. 8E Ch. 17.3 - Prob. 9E Ch. 17.3 - Prob. 10E Ch. 17.3 - Prob. 11E Ch. 17.3 - Prob. 12E Ch. 17.3 - Prob. 13E Ch. 17.3 - Prob. 14E Ch. 17.3 - Prob. 15E Ch. 17.3 - Prob. 16E Ch. 17.3 - Prob. 17E Ch. 17.3 - Prob. 18E Ch. 17.3 - Prob. 19E Ch. 17.3 - Prob. 20E Ch. 17.3 - Prob. 21E Ch. 17.3 - Prob. 22E Ch. 17.3 - Prob. 23E Ch. 17.3 - Figures (a)(f) show the graphs of the functions...Ch. 17.3 - Prob. 25E Ch. 17.3 - Prob. 26E Ch. 17.3 - Prob. 27E Ch. 17.3 - Prob. 28E Ch. 17.3 - Prob. 29E Ch. 17.3 - Prob. 30E Ch. 17.3 - Prob. 31E Ch. 17.3 - Prob. 32E Ch. 17.3 - Prob. 33E Ch. 17.3 - Prob. 34E Ch. 17.3 - Prob. 35E Ch. 17.3 - Prob. 36E Ch. 17.3 - Prob. 37E Ch. 17.3 - Prob. 38E Ch. 17.3 - Prob. 39E Ch. 17.3 - Prob. 40E Ch. 17.3 - Prob. 42E Ch. 17.4 - Prob. 1YT Ch. 17.4 - Prob. 1WE Ch. 17.4 - Prob. 2WE Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Prob. 11E Ch. 17.4 - Find positive numbers x and y such that x + y = 48...Ch. 17.4 - Prob. 13E Ch. 17.4 - Prob. 14E Ch. 17.4 - Find the maximum and minimum values of f(x, y) =x3...Ch. 17.4 - Prob. 16E Ch. 17.4 - Prob. 17E Ch. 17.4 - Prob. 18E Ch. 17.4 - Prob. 19E Ch. 17.4 - Prob. 20E Ch. 17.4 - Prob. 21E Ch. 17.4 - Discuss the advantages and disadvantages of the...Ch. 17.4 - Prob. 23E Ch. 17.4 - Prob. 24E Ch. 17.4 - Prob. 25E Ch. 17.4 - Prob. 26E Ch. 17.4 - Prob. 27E Ch. 17.4 - Prob. 28E Ch. 17.4 - Prob. 29E Ch. 17.4 - Prob. 30E Ch. 17.4 - Prob. 31E Ch. 17.4 - Prob. 32E Ch. 17.4 - Prob. 33E Ch. 17.4 - Prob. 34E Ch. 17.4 - Prob. 35E Ch. 17.4 - Prob. 36E Ch. 17.4 - Prob. 37E Ch. 17.4 - Prob. 38E Ch. 17.4 - Prob. 39E Ch. 17.4 - Prob. 40E Ch. 17.4 - Prob. 41E Ch. 17.4 - Prob. 42E Ch. 17.5 - Prob. 1YT Ch. 17.5 - Prob. 2YT Ch. 17.5 - Prob. 3YT Ch. 17.5 - Prob. 1WE Ch. 17.5 - Prob. 2WE Ch. 17.5 - Prob. 3WE Ch. 17.5 - Prob. 4WE Ch. 17.5 - Prob. 1E Ch. 17.5 - Prob. 2E Ch. 17.5 - Prob. 3E Ch. 17.5 - Prob. 4E Ch. 17.5 - Prob. 5E Ch. 17.5 - Prob. 6E Ch. 17.5 - Prob. 7E Ch. 17.5 - Prob. 8E Ch. 17.5 - Prob. 9E Ch. 17.5 - Prob. 10E Ch. 17.5 - Prob. 11E Ch. 17.5 - Prob. 12E Ch. 17.5 - Prob. 13E Ch. 17.5 - Prob. 14E Ch. 17.5 - Prob. 15E Ch. 17.5 - Prob. 16E Ch. 17.5 - Volume of a Coating An industrial coaling 0.1 in....Ch. 17.5 - Prob. 18E Ch. 17.5 - Prob. 19E Ch. 17.5 - Prob. 20E Ch. 17.5 - Prob. 21E Ch. 17.5 - Prob. 22E Ch. 17.5 - Prob. 23E Ch. 17.5 - Prob. 24E Ch. 17.5 - Life Span As we saw in Exercise 62 of Section...Ch. 17.5 - Prob. 26E Ch. 17.5 - Prob. 27E Ch. 17.5 - Prob. 28E Ch. 17.5 - Prob. 29E Ch. 17.5 - Prob. 30E Ch. 17.5 - Prob. 31E Ch. 17.5 - Prob. 32E Ch. 17.5 - Prob. 33E Ch. 17.5 - Prob. 34E Ch. 17.5 - Prob. 35E Ch. 17.6 - Prob. 1YT Ch. 17.6 - Evaluate 02[13(6x2y2+4xy+8x3+10y4+3)dy]dx and then...Ch. 17.6 - Prob. 3YT Ch. 17.6 - Prob. 4YT Ch. 17.6 - Prob. 5YT Ch. 17.6 - Prob. 1WE Ch. 17.6 - Evaluate each definite integral. 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 amount of steel x tons and aluminum y tons such that to maximize the profit function P ( x , y ) = 36 x y − x 3 − 8 y 3 and use it to find the maximum profit.

Solution Summary: The author calculates the maximum profit of Aunt Mildred's Metal Works, which is 21,600.

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To find: The amount of steel x tons and aluminum y tons such that to maximize the profit function P(x,y)=36xy−x3−8y3 and use it to find the maximum profit.

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Chapter 17 Solutions