Explanation Given: The total revenue (in dollars) of Weston Publishing in publishing and selling English-language dictionaries is, R ( x , y ) = − 0.005 x 2 − 0.003 y 2 − 0.002 x y + 20 x + 15 y (1) Here, x and y denotes the number of deluxe copies and standard copies which were published and sold each day respectively. The total daily cost (in dollars) for publishing is, C ( x , y ) = 6 x + 3 y + 200 (2) Result used: Second derivative test “Let D ( x , y ) = f x x ( x , y ) f y y ( x , y ) − f 2 x y ( x , y ) . Then, check the values of D ( a , b ) as below. (a) If D ( a , b ) > 0 and f x x < 0 then the function f has a relative maximum at the critical point ( a , b ) . (b) If D ( a , b ) > 0 and f x x > 0 then the function f has a relative minimum at the critical point ( a , b ) . (c) If D ( a , b ) < 0 then the function f has a no relative extrema at the critical point ( a , b ) . The point ( a , b , f ( a , b ) ) is said to be a saddle point (d) If D ( a , b ) = 0 then there will be no result. It is to be noted that in (a) and (b) f x x and f y y must have the same sign to satisfy the condition D ( a , b ) > 0 .” Calculation: The contribution to the Weston’s profit from the daily production and the sales of the dictionaries is, P ( x , y ) = R ( x , y ) − C ( x , y ) . Substitute the values of R ( x , y ) and C ( x , y ) from the equation (1) and equation (2). P ( x , y ) = − 0.005 x 2 − 0.003 y 2 − 0.002 x y + 20 x + 15 y − ( 6 x + 3 y + 200 ) = − 0.005 x 2 − 0.003 y 2 − 0.002 x y + 20 x + 15 y − 6 x − 3 y − 200 P ( x , y ) = − 0.005 x 2 − 0.003 y 2 − 0.002 x y + 14 x + 12 y − 200 (3) Obtain the critical point of the function P ( x , y ) . The first derivatives of the function are computed as follows, Partially differentiate the equation (3) with respect to x . ∂ ∂ x P ( x , y ) = ∂ ∂ x ( − 0.005 x 2 − 0.003 y 2 − 0.002 x y + 14 x + 12 y − 200 ) P x = 14 − 0.01 x − 0.002 y (4) Partially differentiate the equation (3) with respect to y . ∂ ∂ y P ( x , y ) = ∂ ∂ y ( − 0.005 x 2 − 0.003 y 2 − 0.002 x y + 14 x + 12 y − 200 ) P y = 12 − 0.002 x − 0.006 y (5) Substitute f x ( x , y ) = 0 and f y ( x , y ) = 0 in the equation (4) and equation (5), Then equation (4) becomes, 14 − 0.01 x − 0.002 y = 0 14000 − 10 x − 2 y = 0 (6) Equation (5) becomes, 12 − 0.002 x − 0.006 y = 0 12000 − 2 x − 6 y = 0 (7) Solve the above system of equations, Multiply the equation (7) by 5, 60 , 000 − 10 x − 30 y = 0 (8) Subtract the equation (6) from the equation (8), 14 , 000 − 10 x − 2 y − 60 , 000 + 10 x + 30 y = 0 − 46 , 000 + 28 y = 0 y ≈ 1642

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 8.3, Problem 22E

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The number of deluxe copies and standard copies that the Weston should publish each day to maximize its profit and find the maximum profit.

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Chapter 8.3, Problem 21E

Chapter 8.3, Problem 23E

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Chapter 8 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

Ch. 8.1 - What is a function of two variables? Give an...Ch. 8.1 - Prob. 2CQ Ch. 8.1 - Prob. 3CQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Let f(x, y) = x2+ 2xy x + 3. Compute f(1, 2),...Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Let h(s, t) = s ln t t ln s. Compute h(1, e),...

Ch. 8.1 - Prob. 8E Ch. 8.1 - Let g(r, s, t) = res/t. Compute g(1, 1, 1), g(1,...Ch. 8.1 - Prob. 10E Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - In Exercises 11-18, find the domain of the...Ch. 8.1 - Prob. 14E Ch. 8.1 - In Exercises 11-18, find the domain of the...Ch. 8.1 - Prob. 16E Ch. 8.1 - In Exercises 11-18, find the domain of the...Ch. 8.1 - Prob. 18E Ch. 8.1 - Prob. 19E Ch. 8.1 - Prob. 20E Ch. 8.1 - In Exercises 19-24, sketch the level curves of the...Ch. 8.1 - Prob. 22E Ch. 8.1 - In Exercises 1924, sketch the level curves of the...Ch. 8.1 - Prob. 24E Ch. 8.1 - Find an equation of the level curve of f (x, y) =...Ch. 8.1 - Prob. 26E Ch. 8.1 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Prob. 30E Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - Prob. 33E Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - POISEITLLES LAW Poiseuilles Law states that the...Ch. 8.1 - COST FUNCTION FOR A LOUDSPEAKER SYSTEM Acrosonic...Ch. 8.1 - Prob. 38E Ch. 8.1 - REVENUE FUNCTIONS FOR DESKS Country Workshop...Ch. 8.1 - Prob. 40E Ch. 8.1 - 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Prob. 30E Ch. 8.5 - Prob. 31E Ch. 8.5 - Prob. 33E Ch. 8.5 - Prob. 34E Ch. 8.5 - Prob. 36E Ch. 8.5 - Prob. 38E Ch. 8.5 - Prob. 39E Ch. 8.5 - Prob. 40E Ch. 8.5 - Prob. 41E Ch. 8.5 - Prob. 42E Ch. 8.6 - Prob. 1CQ Ch. 8.6 - Prob. 2CQ Ch. 8.6 - Prob. 1E Ch. 8.6 - Prob. 2E Ch. 8.6 - Prob. 3E Ch. 8.6 - Prob. 4E Ch. 8.6 - Prob. 5E Ch. 8.6 - Prob. 6E Ch. 8.6 - Prob. 7E Ch. 8.6 - Prob. 8E Ch. 8.6 - Prob. 9E Ch. 8.6 - Prob. 10E Ch. 8.6 - Prob. 11E Ch. 8.6 - Prob. 12E Ch. 8.6 - Prob. 13E Ch. 8.6 - Prob. 14E Ch. 8.6 - In Exercises 318, find the total differential of...Ch. 8.6 - Prob. 16E Ch. 8.6 - Prob. 17E Ch. 8.6 - Prob. 18E Ch. 8.6 - Prob. 19E Ch. 8.6 - Prob. 20E Ch. 8.6 - Prob. 21E Ch. 8.6 - Prob. 22E Ch. 8.6 - Prob. 23E Ch. 8.6 - Prob. 24E Ch. 8.6 - Prob. 25E Ch. 8.6 - Prob. 26E Ch. 8.6 - Prob. 27E Ch. 8.6 - Prob. 28E Ch. 8.6 - Prob. 29E Ch. 8.6 - Prob. 30E Ch. 8.6 - Prob. 31E Ch. 8.6 - Prob. 32E Ch. 8.6 - Prob. 33E Ch. 8.6 - Prob. 34E Ch. 8.6 - Prob. 35E Ch. 8.6 - Prob. 36E Ch. 8.6 - Prob. 37E Ch. 8.6 - 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The number of deluxe copies and standard copies that the Weston should publish each day to maximize its profit and find the maximum profit.

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Chapter 8 Solutions