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Question : Find two positive numbers whose sum is 96 and whose product is a maximum : 2151827

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

Find the indicated relative minimum or maximum.

1) Minimum of f(x,y) = x^{2} + y^{2},

subject to x + y = 1

A) f(0, 1) = (1/2)

B) f(0, 1) = 1

C) f ((1/2), (1/2)) = (1/2)

D) f ((1/2), (1/2)) = 1

2) Minimum of f(x,y) = x^{2} + 2y^{2} - xy,

subject to x + y = 8

A) f(3, 5) = 28

B) f(2, 6) = 25

C) f(6, 2) = 25

D) f(5, 3) = 28

3) Minimum of f(x, y, z) = x^{2} + y^{2} + z^{2},

subject to x + 2y - z = 3

A) f((1/2), 1, - (1/2)) = (3/2)

B) f(- (1/2), 2, (1/2)) = (9/2)

C) f(0, 1, -1) = 2

D) f((3/7), (6/7), - (3/7)) = (54/49)

4) Maximum of f(x,y) = 4xy,

subject to x + y = 8

A) f(3, 5) = 64

B) f(2, 6) = 72

C) f(4, 4) = 64

D) f(0, 8) = 72

5) Minimum of f(x,y) = x^{2} - 14x + y^{2} - 16y,

subject to 2x + 3y = 12

A) f(3, 2) = -61

B) f(1, 5) = -68

C) f(0, 1) = -15

D) f(2, 0) = -24

6) Maximum of f(x,y) = xy,

subject to x + y = 100

A) f(100, 0) = 0

B) f(50, 50) = 2500

C) f(50, 50) = 100

D) f(0, 100) = 0

7) Minimum of f(x,y) = x^{2} + y^{2} - xy,

subject to x - y = 10

A) f(5, 5) = 25

B) f(2, -1) = 7

C) f(5, -5) = 75

D) f(1, 2) = 3

8) Minimum of f(x,y) = x^{2} + 4y^{2} + 6,

subject to 2x - 8y = 20

A) f(2, -2) = 26

B) f(-2, 2) = 26

C) f(-2, -2) = 26

D) f(2, 2) = 26

9) Minimum of f(x, y) = x^{2} + y^{2},

subject to x - 3y = 6

A) f((9/5), - (7/5)) = (26/5)

B) f(- (9/5), - (13/5)) = 10

C) f(- (3/5), - (11/5)) = (26/5)

D) f((3/5), - (9/5)) = (18/5)

10) Maximum of f(x, y, z) = xy + z,

subject to x^{2} + y^{2} + z^{2} = 1

A) f(0, 0, 1) = 1

B) f(1, 1, 0) = 1

C) f(0, 1, 0) = 1

D) f(1, 1, 1) = 1

Solve the problem.

11) Find two positive numbers whose sum is 96 and whose product is a maximum.

A) 72 and 72

B) 48 and 72

C) 48 and 48

D) 24 and 72

12) 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 = 15 and y = 45

C) x = 30 and y = 30

D) x = 1 and y = 59

13) Find three positive numbers whose sum is 84 and whose product is a maximum.

A) 28, 21, and 21

B) 42, 42, and 42

C) 28, 28, and 28

D) 42, 21, and 21

14) The total cost to hand-produce x large dolls and y small dolls is given by C(x,y) = 2x^{2} + 5y^{2} + 4xy + 70. If a total of 70 dolls must be made, how should production be allocated so that the total cost is minimized?

A) Make 35 large dolls and 35 small ones

B) Make 69 large dolls and 1 small one

C) Make 70 large dolls and 0 small ones

D) Make 0 large dolls and 70 small ones

15) 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 $2500/unit, capital costs $700/unit and the owner has $1,700,000 available for one time period, what amount of labor and capital would maximize production?

A) 1214.3 units of labor and 340.0 units of capital

B) 340.0 units of labor and 1214.3 units of capital

C) 326.9 units of labor and 1062.5 units of capital

D) 680.0 units of labor and 2428.6 units of capital

16) A farmer has 260 m of fencing. Find the dimensions of the rectangular field of maximum area that can be enclosed by this amount of fencing.

A) 55 m by 75 m

B) 65 m by 195 m

C) 26 m by 104 m

D) 65 m by 65 m

17) A farmer has 200 m of fencing. Find the area of the largest rectangular field that he can enclose with his fencing. Assume that no fencing is needed along one edge of the field.

A) 5000 m^{2}

B) 6525 m^{2}

C) 10,000 m^{2}

D) 16,275 m^{2}

18) What are the dimensions of a rectangular box, open at the top, which has maximum volume when the surface area is 48 in^{2}?

A) x = 8 in, y = 2 in, z = 6 in

B) x = 6 in, y = 6 in, z = 3 in

C) x = 4 in, y = 4 in, z = 2 in

D) x = 4 in, y = 2 in, z = 2 in

19) The material for the bottom of a rectangular box costs $3 per square foot while the material for the sides and top costs $1 per square foot. Find the greatest capacity such a box can have if the total amount available for material is $12.

A) 3 ft^{3}

B) 2 ft^{3}

C) 4 ft^{3}

D) 1 ft^{3}

20) Find the dimensions of the right circular cylinder with maximum volume if its surface area is 24π in^{2}.

A) r = 2 in, h = 6 in

B) r = 2 in, h = 4 in

C) r = 3 in, h = 3 in

D) r = 3 in, h = 8 in

21) Find the dimensions of the right circular cylinder with maximum surface area, if its volume is 64 ft^{3}.

A) r = 6 ft, h = 1 ft

B) r = 8 ft, h = (π/16) ft

C) r = 3 ft, h = 2 ft

D) r = √(2) ft, h = (32/π) ft

22) What is the greatest area that a rectangle can have if the length of its diagonal is 2 m?

A) 1 m^{2}

B) 2 m^{2}

C) 5 m^{2}

D) 2√(2) m^{2}

23) 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 110 centimeters and radius 330/π centimeters

B) Height 330 centimeters and radius 110/π centimeters

C) Height 220 centimeters and radius 110 centimeters

D) Height 110 centimeters and radius 110/π centimeters

24) A rectangular box with square base and no top is to have a volume of 32 ft^{3}. What is the least amount of material required?

A) 36 ft^{2}

B) 40 ft^{2}

C) 48 ft^{2}

D) 42 ft^{2}