Explanation Given: The both length and the girth together of the rectangular package does not exceed 130 in. That is, 2 x + 2 z + 2 y = 130 (1) The volume of the rectangular package is x y z . Result used: 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 . Calculation: Let the dimensions of the rectangular package be x × y × z . Multiply the equation (1) by xz to obtain the volume function. x ( 2 x + 2 z + y ) = 130 x z 2 x 2 z + 2 x z 2 + 2 x y z = 130 x z 2 x 2 z + 2 x z 2 + 2 V = 130 x z V = 65 x z − x 2 z − x z 2 Thus, the function is f ( x , z ) = 65 x z − x 2 z − x z 2 (2) The first derivatives of the volume function are computed as follows, Partially differentiate the equation (2) with respect to x . ∂ ∂ x f ( x , z ) = ∂ ∂ x ( 65 x z − x 2 z − x z 2 ) = ∂ ∂ x ( 65 x z ) − ∂ ∂ x ( x 2 z ) − ∂ ∂ x ( x z 2 ) = 65 z − 2 x z − z 2 Thus, f x = 65 z − 2 x z − z 2 (3) Partially differentiate the equation (2) with respect to y . ∂ ∂ z f ( x , z ) = ∂ ∂ z ( 65 x z − x 2 z − x z 2 ) = ∂ ∂ z ( 65 x z ) − ∂ ∂ z ( x 2 z ) − ∂ ∂ z ( x z 2 ) = 65 x − x 2 − 2 x z Hence, f z = 65 x − x 2 − x z (4) Substitute f x ( x , y ) = 0 and f z ( x , y ) = 0 in the equation (3) and equation (4), Then equation (6) becomes, ( 65 − 2 x − z ) = 0

Applied Calculus for the Managerial, Life, and Social Sciences

10th Edition

ISBN: 9781305887831

Author: Tan

Publisher: CENGAGE CO

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Cylinders

A cylinder is a three-dimensional solid shape with two parallel and congruent circular bases, joined by a curved surface at a fixed distance. A cylinder has an infinite curvilinear surface.

Cones

A cone is a three-dimensional solid shape having a flat base and a pointed edge at the top. The flat base of the cone tapers smoothly to form the pointed edge known as the apex. The flat base of the cone can either be circular or elliptical. A cone is dr…

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Chapter 8.3, Problem 33E

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The dimensions of the rectangular package.

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Chapter 8.3, Problem 32E

Chapter 8.3, Problem 34E

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Applied Calculus for the Managerial, Life, and Social Sciences

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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