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Question : Find the pairs (x, y) that give the extreme values of 2x + 10y, subject to the constraint 4x^2 + 5y^2 = 8400 : 2163364

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

35) Find the pairs (x, y) that give the extreme values of 2x + 10y, subject to the constraint 4x^{2} + 5y^{2} = 8400, using the method of Lagrange multipliers.

Enter your answer as exactly just (a, b), (c, d) where a > c (no words).

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

36) Suppose the partial derivatives of a Lagrange function F(x, y, λ) are (∂F/∂x) = 2 - 8λx, (∂F/∂y) = 1 -2λy, (∂F/∂λ) = 32 - 4x^{2} - y^{2}. What values of x and y minimize F(x, y, λ)? (Assume x and y are positive.)

A) (4, 2)

B) (2√(2), √(4))

C) (2, 4)

D) (2, √(2))

E) none of these

37) Which of the following pairs of values (x, y) maximizes the function f(x, y) = x + 3y subject to the constraint x^{2} + 9y^{2} = 72 assuming x and y are positive?

(I) (-6, -2) (II) (8, √((8/9))) (III) (2, -6) (IV) (6, 2)

A) I and II

B) IV only

C) I only

D) III only

E) none of these

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

38) Maximize f(x, y, z) = x + y + z subject to the constraint x^{2} + y^{2} + z^{2} = 1, x, y, z > 0.

Enter your answer as just 3^{a} where a is a reduced fraction of form (b/c).

39) Minimize the function f(x, y, z) = x^{2} + y^{2} + z^{2} subject to the constraint x + y + z = 2.

Enter your answer an just a reduced fraction of form (a/b).

40) Maximize the function f(x, y) = x^{2} - y^{2} subject to the constraint y - x^{2} = - (1/2), x, y >0.

Enter your answer as just a reduced fraction of form (a/b).

41) Maximize the function f(x, y) = e^{xy} subject to the constraint x^{2} + y^{2} = 18, x, y > 0.

Enter your answer as just e^{a}.

42) Maximize the function f(x, y) = 2x + y subject to the constraint x^{2} + y^{2} = 1.

Enter your answer exactly in the form (a√((b/c)), √((d/e))) , (-h√((i/j)), √((k/l))).

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the problem.

43) Find two positive numbers x and y such that x + y = 60 and xy^{2} is maximized.

A) x = 20 and y = 40

B) x = 1 and y = 59

C) x = 30 and y = 30

D) x = 15 and y = 45

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

44) A business produces two products A and B. Let x and y denote, respectively, the quantity of A and B to be produced. Limitations on the company resources require that 500x^{2} + 100y be at most 100,000. Each unit of A yields a $5000 profit and each unit of B yields a $500 profit. What should x and y be to yield a maximum profit?

Enter your answer exactly as just (a, b) an ordered pair of integers.

45) Design a cylindrical can of volume 100 cubic units that requires a minimum amount of aluminum; that is, the can is to have a minimum surface area.

Enter your answer exactly as just r, h where r is exactly of form ((3)√(a/b)) representing radius, and h is exactly of form c((3)√(d/e)) representing height (no labels, words, or units).

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

46) A closed rectangular box with square ends is to be designed so that the surface area of the box is minimized. [Note: Surface area = 2x^{2} + 4xy.] It is required that the volume be 32 cubic inches. Which of the following is the Lagrange function F(x, y, λ) for this problem?

A) (x^{2}y - 32) + λ(2x^{2} + 4xy)

B) 2x^{2} + 4xy + λ(32)

C) 2x^{2} + 4xy + λ(x^{2}y - 32)

D) x^{2}y + λ(2x^{2} + 4xy - 32)

E) none of these

47) An artist produces two items for sale. Each unit of item I costs $50 to produce, while each unit of item II costs $200. The revenue function is R(x, y) = 40x + 7xy + 80y^{2} + 10y, where x is units of item I and y is units of item II. Suppose the artist has only $1000 to spend on production. Which of the following is the Lagrange function the artist should use to determine what combination of production amounts (x, y) will yield maximum profits subject to the constraint that his costs must equal $1000.

A) 40x + 7xy + 80y^{2} + 10y + λ(200x + 50y - 1000)

B) -10x + 7xy + 80y^{2} - 190y + λ(200x + 50y - 1000)

C) -10x + 7xy + 80y^{2} - 190y + λ(50x + 200y - 1000)

D) 40x + 7xy + 80y^{2} + 10y + λ(50x + 200y - 1000)

E) none of these

Solve the problem.

48) Find two positive numbers whose sum is 64 and whose product is a maximum.

A) 32 and 32

B) 16 and 48

C) 48 and 48

D) 32 and 48

49) Find three positive numbers whose sum is 108 and whose product is a maximum.

A) 36, 36, and 36

B) 36, 27, and 27

C) 54, 54, and 54

D) 54, 27, and 27

50) The production level P of a factory during one time period is modeled by P(x, y) = Kx^{1/2}y^{1/2} where K is a positive integer, x is the number of units of labor scheduled and y is the number of units of capital invested. If labor costs $4500/unit, capital costs $700/unit and the owner has $1,900,000 available for one time period, what amount of labor and capital would maximize production?

A) 1357.1 units of labor and 211.1 units of capital

B) 422.2 units of labor and 2714.3 units of capital

C) 206.5 units of labor and 1187.5 units of capital

D) 211.1 units of labor and 1357.1 units of capital

51) A farmer has 780 m of fencing. Find the dimensions of the rectangular field of maximum area that can be enclosed by this amount of fencing.

A) 185 m by 205 m

B) 195 m by 195 m

C) 195 m by 585 m

D) 78 m by 312 m

52) Assuming that a cylindrical container can be mailed only if the sum of its height and circumference do not exceed 330 centimeters, what are the dimensions of the cylinder with the largest volume that can be mailed?

A) Height 220 centimeters and radius 110 centimeters

B) Height 330 centimeters and radius 110/π centimeters

C) Height 110 centimeters and radius 330/π centimeters

D) Height 110 centimeters and radius 110/π centimeters