Explanation Given Information: The dimensions of a closed-top box are given by length x , width y , and height z (in inches) and the volume of the box is 27 , 000 cubic inches. Formula used: To calculate relative maxima and minima of the function z = f ( x , y ) , (1) Find the partial derivatives ∂ z ∂ x and ∂ z ∂ y . (2) Find the critical points, that is, the point(s) that satisfy ∂ z ∂ x = 0 and ∂ z ∂ y = 0 . (3) Then find all the second partial derivatives and evaluate the value of D at each critical point, where D = ( z x x ) ( z y y ) − ( z x y ) 2 = ∂ 2 z ∂ x 2 ⋅ ∂ 2 z ∂ y 2 − ( ∂ 2 z ∂ x ∂ y ) 2 . (a) If D > 0 , then relative minimum occurs if z x x > 0 and relative maximum occurs if z x x < 0 . (b) If D < 0 , then neither a relative maximum nor a relative minimum occurs. For a function f ( x , y ) , the partial derivative of f with respect to x is calculated by taking the derivative of f ( x , y ) with respect to x and keeping the other variable y constant and the partial derivative of f with respect to y is calculated by taking the derivative of f ( x , y ) with respect to y and keeping the other variable x constant. The partial derivative of f with respect to x is denoted by f x and with respect to y is denoted by f y . For a function z ( x , y ) , the second partial derivative, (1) When both derivatives are taken with respect to x is z x x = ∂ 2 z ∂ x 2 = ∂ ∂ x ( ∂ z ∂ x ) . (2) When both derivatives are taken with respect to y is z y y = ∂ 2 z ∂ y 2 = ∂ ∂ y ( ∂ z ∂ y ) . (3) When first derivative is taken with respect to x and second derivative is taken with respect to y is z x y = ∂ 2 z ∂ y ∂ x = ∂ ∂ y ( ∂ z ∂ x ) . (4) When first derivative is taken with respect to y and second derivative is taken with respect to x is z y x = ∂ 2 z ∂ x ∂ y = ∂ ∂ x ( ∂ z ∂ y ) . Power of x rule for a real number n is such that, if f ( x ) = x n then f ′ ( x ) = n x n − 1 . Quotient rule for function f ( x ) = u ( x ) v ( x ) , where u and v are differentiable functions of x , then f ′ ( x ) = v ( x ) ⋅ u ′ ( x ) − u ( x ) ⋅ v ′ ( x ) ( v ( x ) ) 2 . Constant function rule for a constant c is such that, if f ( x ) = c then f ′ ( x ) = 0 . Coefficient rule for a constant c is such that, if f ( x ) = c ⋅ u ( x ) , where u ( x ) is a differentiable function of x , then f ′ ( x ) = c ⋅ u ′ ( x ) . The surface area of a box with closed top and length l , width w , and height h is A = 2 l h + 2 w h + 2 l w . The volume of a box with length l , width w , and height h is l w h . Calculation: Consider the problem, the dimensions of a closed-top box are given by length x , width y , and height z (in inches) and the volume of the box is 27 , 000 cubic inches. Use the formula for the surface area of a box with closed top and length l , width w , and height h . A = 2 l h + 2 w h + 2 l w . Substitute x for l , y for w , and z for h in A = 2 l h + 2 w h + 2 l w . A = 2 x z + 2 y z + 2 x y Use the formula for the volume of a box with length l , width w , and height h . V = l w h Substitute x for l , y for w , and z for h in V = l w h . V = x y z Since, the volume of the box is 27 , 000 cubic inches. Substitute 27 , 000 for V . 27 , 000 = x y z z = 27 , 000 x y Substitute 27 , 000 x y for z in A = 2 x z + 2 y z + 2 x y . A = 2 x ( 27 , 000 x y ) + 2 y ( 27 , 000 x y ) + 2 x y = 54 , 000 y + 54 , 000 x + 2 x y Thus, the area function is A ( x , y ) = 54 , 000 y + 54 , 000 x + 2 x y . Differentiate A = f ( x , y ) with respect to x by holding y constant, ∂ A ∂ x = ∂ ∂ x ( 54 , 000 y + 54 , 000 x + 2 x y ) = − 54 , 000 x 2 + 2 y Equate the first derivative of function A ( x , y ) = 54 , 000 y + 54 , 000 x + 2 x y with respect to x to zero. − 54 , 000 x 2 + 2 y = 0 2 y = 54 , 000 x 2 y = 27 , 000 x 2 The first partial derivative of function A ( x , y ) = 54 , 000 y + 54 , 000 x + 2 x y with respect to y by holding x constant. ∂ A ∂ y = ∂ ∂ y ( 54 , 000 y + 54 , 000 x + 2 x y ) = − 54 , 000 y 2 + 2 x Equate the first derivative of function A ( x , y ) = 54 , 000 y + 54 , 000 x + 2 x y with respect to y to zero

Mathematical Applications for the Management, Life, and Social Sciences

12th Edition

ISBN: 9781337625340

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

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Chapter 14.4, Problem 28E

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To calculate: The values of each of the dimensions of a closed-top box of length x, width y, and height z (in inches) if the volume equals 27,000 cubic inches and the box requires the least amount of material to make.

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Chapter 14.4, Problem 27E

Chapter 14.4, Problem 29E

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Chapter 14 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 14.1 - CHECKPOINT 1. Find the domain of the function Ch. 14.1 - CHECKPOINT 2. (a) If . (b) If . Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E Ch. 14.1 - Prob. 6E Ch. 14.1 - Prob. 7E Ch. 14.1 - Prob. 8E

Ch. 14.1 - Prob. 9E Ch. 14.1 - Prob. 10E Ch. 14.1 - Prob. 11E Ch. 14.1 - Prob. 12E Ch. 14.1 - Prob. 13E Ch. 14.1 - Prob. 14E Ch. 14.1 - Prob. 15E Ch. 14.1 - Prob. 16E Ch. 14.1 - In Problems 15-22, evaluate each function as...Ch. 14.1 - Prob. 18E Ch. 14.1 - Prob. 19E Ch. 14.1 - Prob. 20E Ch. 14.1 - Prob. 21E Ch. 14.1 - In Problems 15-22, evaluate each function as...Ch. 14.1 - Prob. 23E Ch. 14.1 - Prob. 24E Ch. 14.1 - 25. Curve speeds One method traffic planners use...Ch. 14.1 - Prob. 26E Ch. 14.1 - Prob. 27E Ch. 14.1 - Prob. 28E Ch. 14.1 - 29. Mortgage The following tables show that a...Ch. 14.1 - 30. Wind chill Wind and cold temperatures combine...Ch. 14.1 - Prob. 31E Ch. 14.1 - Prob. 32E Ch. 14.1 - Prob. 33E Ch. 14.1 - Prob. 34E Ch. 14.1 - Prob. 35E Ch. 14.1 - 36. Profit The Kirk Kelly Kandy Company makes two...Ch. 14.1 - Prob. 37E Ch. 14.1 - Prob. 38E 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. 1E Ch. 14.2 - Prob. 2E 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 - 19. Find the slope of the tangent in the...Ch. 14.2 - 20. Find the slope of the tangent in the...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 - 32. If , find the following. (a) (b) (c) (d) Ch. 14.2 - 33. If , find the following. Ch. 14.2 - 34. If , find the following. Ch. 14.2 - Prob. 35E Ch. 14.2 - Ch. 14.2 - Ch. 14.2 - Prob. 38E Ch. 14.2 - Prob. 39E Ch. 14.2 - Prob. 40E Ch. 14.2 - 41. . Ch. 14.2 - . Ch. 14.2 - Prob. 43E Ch. 14.2 - Ch. 14.2 - 45. Mortgage When a homeowner has a 25-year...Ch. 14.2 - 46. Mass transportation ridership Suppose that in...Ch. 14.2 - 47. Wilson's lot size formula In economics, the...Ch. 14.2 - 48. Cost Suppose that the total cost (in dollars)...Ch. 14.2 - 49. Pesticide Suppose that the number of thousands...Ch. 14.2 - 50. Profit Suppose that the profit (in dollars)...Ch. 14.2 - Prob. 51E Ch. 14.2 - Prob. 52E Ch. 14.2 - 53. Production Suppose that the output Q (in...Ch. 14.2 - Prob. 54E Ch. 14.2 - Prob. 55E Ch. 14.2 - Prob. 56E Ch. 14.3 - CHECKPOINT If the joint cost in dollars for two...Ch. 14.3 - Prob. 2CP Ch. 14.3 - Prob. 3CP Ch. 14.3 - Prob. 1E Ch. 14.3 - Prob. 2E Ch. 14.3 - 3. The total cost of producing 1 unit of a product...Ch. 14.3 - Prob. 4E Ch. 14.3 - Prob. 5E Ch. 14.3 - Prob. 6E Ch. 14.3 - Prob. 7E Ch. 14.3 - Prob. 8E Ch. 14.3 - 9. If the joint cost function for two products is ...Ch. 14.3 - 10. Suppose the joint cost function for x units of...Ch. 14.3 - 11. Suppose that the joint cost function for two...Ch. 14.3 - Prob. 12E Ch. 14.3 - Prob. 13E Ch. 14.3 - Prob. 14E Ch. 14.3 - Prob. 15E Ch. 14.3 - Prob. 16E Ch. 14.3 - Prob. 17E Ch. 14.3 - Prob. 18E Ch. 14.3 - Prob. 19E Ch. 14.3 - Prob. 20E Ch. 14.3 - 21. Suppose the Cobb-Douglas production function...Ch. 14.3 - Prob. 22E Ch. 14.3 - Prob. 23E Ch. 14.3 - Prob. 24E Ch. 14.3 - Prob. 25E Ch. 14.3 - Prob. 26E Ch. 14.3 - Prob. 27E Ch. 14.3 - Prob. 28E Ch. 14.3 - Prob. 29E Ch. 14.3 - Prob. 30E Ch. 14.4 - CHECKPOINT Suppose that Find Ch. 14.4 - Prob. 2CP Ch. 14.4 - Prob. 3CP Ch. 14.4 - Prob. 4CP Ch. 14.4 - Prob. 1E Ch. 14.4 - Prob. 2E Ch. 14.4 - Prob. 3E Ch. 14.4 - Prob. 4E Ch. 14.4 - Prob. 5E Ch. 14.4 - Prob. 6E Ch. 14.4 - Prob. 7E Ch. 14.4 - Prob. 8E Ch. 14.4 - Prob. 9E Ch. 14.4 - Prob. 10E Ch. 14.4 - Prob. 11E Ch. 14.4 - Prob. 12E Ch. 14.4 - Prob. 13E Ch. 14.4 - Prob. 14E Ch. 14.4 - Prob. 15E Ch. 14.4 - Prob. 16E Ch. 14.4 - Prob. 17E Ch. 14.4 - Prob. 18E Ch. 14.4 - 19. Profit Suppose that the quarterly profit from...Ch. 14.4 - Prob. 20E Ch. 14.4 - 21. Nutrition A new food is designed to add weight...Ch. 14.4 - Prob. 22E Ch. 14.4 - 23. Production Suppose that tons is the...Ch. 14.4 - 24. Production Suppose that x units of one input...Ch. 14.4 - 25. Profit Suppose that a manufacturer produces...Ch. 14.4 - Prob. 26E Ch. 14.4 - Prob. 27E Ch. 14.4 - Prob. 28E Ch. 14.4 - 29. Profit A company manufactures two products, A...Ch. 14.4 - Prob. 30E Ch. 14.4 - Prob. 31E Ch. 14.4 - The manager of the Sea Islands Chicken Shack is...Ch. 14.4 - Prob. 33E Ch. 14.4 - Prob. 34E Ch. 14.4 - Prob. 35E Ch. 14.4 - Prob. 36E Ch. 14.5 - Prob. 1CP Ch. 14.5 - Prob. 2CP Ch. 14.5 - Prob. 3CP Ch. 14.5 - Prob. 4CP Ch. 14.5 - Prob. 1E Ch. 14.5 - Prob. 2E Ch. 14.5 - Prob. 3E Ch. 14.5 - Prob. 4E Ch. 14.5 - Prob. 5E Ch. 14.5 - Prob. 6E Ch. 14.5 - Prob. 7E Ch. 14.5 - Prob. 8E Ch. 14.5 - Prob. 9E Ch. 14.5 - Prob. 10E Ch. 14.5 - Prob. 11E Ch. 14.5 - Prob. 12E Ch. 14.5 - Prob. 13E Ch. 14.5 - Prob. 14E Ch. 14.5 - Prob. 15E Ch. 14.5 - 16. Utility Suppose that the budget constraint in...Ch. 14.5 - 17. Utility Suppose that the utility function for...Ch. 14.5 - 18. Utility Suppose that the utility function for...Ch. 14.5 - Prob. 19E Ch. 14.5 - Prob. 20E Ch. 14.5 - 21. Cost A firm has two plants, X and Y. Suppose...Ch. 14.5 - Prob. 22E Ch. 14.5 - Prob. 23E Ch. 14.5 - Prob. 24E Ch. 14.5 - 25. Manufacturing Find the dimensions (in...Ch. 14.5 - Prob. 26E Ch. 14 - 1. What is the domain of ? Ch. 14 - Prob. 2RE Ch. 14 - Prob. 3RE Ch. 14 - 4. If . Ch. 14 - Prob. 5RE Ch. 14 - Prob. 6RE Ch. 14 - Prob. 7RE Ch. 14 - Prob. 8RE Ch. 14 - Prob. 9RE Ch. 14 - Prob. 10RE Ch. 14 - Prob. 11RE Ch. 14 - Prob. 12RE Ch. 14 - Prob. 13RE Ch. 14 - Prob. 14RE Ch. 14 - In Problems 15-18, find the second partials. Ch. 14 - Prob. 16RE Ch. 14 - Prob. 17RE Ch. 14 - Prob. 18RE Ch. 14 - Prob. 19RE Ch. 14 - Prob. 20RE Ch. 14 - Prob. 21RE Ch. 14 - Prob. 22RE Ch. 14 - Prob. 23RE Ch. 14 - Prob. 24RE Ch. 14 - Prob. 25RE Ch. 14 - Prob. 26RE Ch. 14 - Prob. 27RE Ch. 14 - Prob. 28RE Ch. 14 - Prob. 29RE Ch. 14 - Prob. 30RE Ch. 14 - Prob. 31RE Ch. 14 - Prob. 32RE Ch. 14 - Prob. 33RE Ch. 14 - Prob. 34RE Ch. 14 - 35. Modeling US. average wage The table gives the...Ch. 14 - Prob. 36RE Ch. 14 - Prob. 1T Ch. 14 - Prob. 2T Ch. 14 - Prob. 3T Ch. 14 - Prob. 4T Ch. 14 - Prob. 5T Ch. 14 - Prob. 6T Ch. 14 - 7. Suppose the demand functions for two products...Ch. 14 - Prob. 8T Ch. 14 - 9. Find x and y that maximize the utility function...Ch. 14 - Prob. 10T

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The values of each of the dimensions of a closed-top box of length x , width y , and height z (in inches) if the volume equals 27 , 000 cubic inches and the box requires the least amount of material to make.

Solution Summary: The author calculates the dimensions of a closed-top box if the volume equals 27,000 cubic inches and the box requires the least amount of material to make.

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To calculate: The values of each of the dimensions of a closed-top box of length x, width y, and height z (in inches) if the volume equals 27,000 cubic inches and the box requires the least amount of material to make.

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Chapter 14 Solutions