Explanation Given: The volume of the rectangular box shaped building is, x y z = 12000 ft 3 (1) The annual cost for the cooling and heating the top, front and back is $ 2 / f t 2 and $ 4 / f t 2 respectively, and for the sides is $ 3 / f t 2 . Result used: (1) The critical point of the function ( a , b ) in the domain f is obtained by solving the below inequalities, f y ( x , y ) = 0 and f x ( x , y ) = 0 . (2) 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: Let the dimensions of the open rectangular box be x × y × z . The surface area of the box as follows, S = 2 ( x y + y z + x z ) = 2 ( 2 x y + 4 y z + 3 x z ) S = 4 x y + 8 y z + 3 x z Let the surface area be the function f ( x , y ) = 4 x y + 8 y z + 3 x z (2) From the equation (1), obtain the value of yz , x y z = 12 , 000 y z = 12 , 000 x (3) Obtain the value of xz , x z = 12 , 000 y (4) Substitute the values y z = 12 , 000 x and x z = 12 , 000 y in the equation (2). S = 4 x y + 8 ( 12000 x ) + 3 ( 12000 y ) f ( x , y ) = 4 x y + ( 96000 x ) + ( 36000 y ) (5) Obtain the critical point of the function f ( x , y ) . The first derivatives of the function are computed as follows, Partially differentiate the equation (5) with respect to x . ∂ ∂ x f ( x , y ) = ∂ ∂ x ( 4 x y + ( 96000 x ) + ( 36000 y ) ) f x = 4 y − 96000 x 2 (6) Partially differentiate the equation (5) with respect to y . ∂ ∂ y f ( x , y ) = ∂ ∂ y ( 4 x y + ( 96000 x ) + ( 36000 y ) ) f y = 4 x − 36000 y 2 (7) Substitute f x ( x , y ) = 0 and f y ( x , y ) = 0 in the equation (6) and equation (7), Then equation (6) becomes, 4 y − 96000 x 2 = 0 y = 24000 x 2 (8) The equation (7) becomes, 4 x − 36000 y 2 = 0 Substitute the value y = 24000 x 2 in 4 x − 36000 y 2 = 0 , 4 x − 36000 ( 24000 x 2 ) 2 = 0 ( 24000 x 2 ) 2 x − 36000 = 0 36000 x 3 = 2304000000 x 3 = 64000 That is, x = 40

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 34E

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Chapter 8.3, Problem 33E

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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),...

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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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