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Question : The width of a rectangle is measured as 12.2 cm, while the length is measured : 2151832

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

Evaluate dz.

1) z = -4x^{2} + 6xy + 2y^{2};

x = 5, y = 4, dx = 0.02, dy = -0.03

A) -1.70

B) 1.70

C) 1.40

D) -1.40

2) z = (2x/y) + √(17xy);

x = 10, y = 5, dx = 0.03, dy = 0.03

A) 0.42

B) -0.12

C) 0.12

D) -0.42

3) z = ln(5x + 2y);

x = 4, y = 12, dx = 0.03, dy = -0.02

A) 0.003

B) -0.003

C) -0.001

D) 0.001

4) z = x^{3} - 3xy^{2} + 3y; x = -3, y = -3, dx = -0.03, dy = 0.02

A) -1.08

B) -2.37

C) 0.42

D) -1.02

5) z = (x^{2} + 6y/3x^{2} - y); x = -1, y = 2, dx = 0.02, dy = -0.02

A) -0.06000

B) 1.38000

C) -1.74000

D) 1.14000

Evaluate dw.

6) w = 3x^{3}√(2y)ln(8z);

x = 9, y = 2, z = 5,

dx = 0.02, dy = -0.02, dz = 0.01

A) 26.89

B) 116.32

C) -71.93

D) 35.64

7) w = 5x^{2}y + 4x^{4}z;

x = 6, y = 4, z = 6,

dx = -0.03, dy = 0.02, dz = 0.03

A) -473.76

B) -625.68

C) -470.16

D) 151.92

Use the total differential to approximate the quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference of the two results to four decimal places.

8) √(1.02^{2} + 3.98^{2})

A) 4.1086; 4.1086; 0

B) 3.1086; 4.1086; 1.0000

C) 4; 4.1086; 0.1086

D) 3.150; 3.1590; 0.0090

9) (5.01^{2} - 0.93^{2})^{1/3}

A) 2; 2.8939; 0.8939

B) 2.8941; 2.8939; 0.0002

C) 1.8941; 2.8939; 1.0002

D) 2.8941; 2; 0.8941

10) 0.97e^{0.02}

A) 1.01; 0.9896; 0.0204

B) 0.99; 1; 0.0100

C) 1; 0.9896; 0.0104

D) 0.99; 0.9896; 0.0004

11) 0.98ln(1.02)

A) 0.02; 0.0194; 0.0006

B) 1; .02; 0.9800

C) 0.02; 0.01; 0.0100

D) 0; 0.0194; 0.0194

Solve the problem.

12) The cost to make a product is M(x,y) = 10x^{2} + 3y^{2} - 11xy + 50, where x is material cost, y is labor cost. The company spends $14 on materials and $12 on labor. Use the differential to estimate the change in cost if the company spends $19 on materials and $10 on labor.

A) $1480

B) $904

C) -$328

D) -$1480

13) The production function for a certain country is z = x^{0}.3y^{0}.4, where x stands for units of labor and y for units of capital. At present, x is 29 and y is 46. Use differentials to estimate the change in z if x becomes 34 and y becomes 52.

A) 1.21

B) 1.32

C) 1.34

D) 1.45

14) The width of a rectangle is measured as 12.2 cm, while the length is measured as 23.1 cm. The width measurement could be off by 0.9 cm, and the length could be off by 0.7 cm. Estimate the maximum possible error in calculating the area of the rectangle.

A) 27.15 cm^{2}

B) 17.08 cm^{2}

C) 41.58 cm^{2}

D) 29.33 cm^{2}

15) Approximate the amount of aluminum needed for a beverage can of radius 2.8 cm and height 16 cm. Assume the walls of the can are 0.1 cm thick.

A) 330.75 cm^{3}

B) 28.15 cm^{3}

C) 4.93 cm^{3}

D) 33.07 cm^{3}

16) The number of liters of blood pumped through the lungs in one minute is given by

C = (b/a - v)

Suppose a = 150, b = 180, v = 120. Estimate the change in C if a becomes 140, b becomes 186, and v changes to 128.

A) 0.6 liters

B) 3.4 liters

C) 3.8 liters

D) -0.2 liters

17) The dimensions of a cardboard box were measured as 30 cm, 70 cm and 50 cm, with the percentage error in each dimension not exceeding 5 percent. Approximate the worst possible percentage error in the volume of the box. Assume that the cardboard is of negligible thickness.

A) 5%

B) 10%

C) 11%

D) 15%

18) The base radius and the height of a cylindrical can tank were measured at 25 cm and 55 cm.The percentage errors in these measurements do not exceed 0.2 for the base radius and 0.7 for the height. Approximate the worst possible percentage error in the volume of the can. Assume that the sides and ends of the can are of negligible thickness.

A) 1.8%

B) 0.9%

C) 1.1%

D) 1.4%

19) The storage tank has the form of a cylinder with one hemisphere end (see figure). The perimeter of a circular cross-section perpendicular to the axis of symmetry of the tank was measured at 31.4 ± 3 ft and the perimeter of a cross-section lying in the plane containing the axis of symmetry was measured at 35.7 ± 1.5 ft. Approximate the worst possible percentage error in the volume of the tank. Assume that the sides and ends of the tank are of negligible thickness.

A) 1191.2

B) 0.3

C) 26.5

D) 238.2

20) The resistance R of a circuit containing two resistors in parallel is (1/R) = (1/R_{1}) + (1/R_{2}), where the two resistors have resistances R_{1} and R_{2}. For resistors with R_{1} = 200 ± 0.4 ohm and R_{2} = 110 ± 0.8 ohm, approximate the error in R.

A) 0.4

B) 0.9

C) 1.1

D) -0.1