Deck 7: Applications of Trigonometry

Question 1

Find (a) the dot product of the two vectors and (b) the angle between the two vectors. and

A)
B)
C)
D)
E)

Answer 1

C

Question 2

Given that , find the number.

A) 13
B) 10
C) -7
D) 25
E) 5

Answer 2

C

Question 3

Shown in the figure is a plan for the top of a wing of a jet fighter, where feet and feet. Approximate angle .

A)
B)
C)
D)
E)

Answer 3

D

Question 4

Shown in the figure is a plan for the top of a wing of a jet fighter, where feet and feet. Approximate the area of triangle ABC.

A) 453.3 ft ²
B) 452.9 ft ²
C) 905.7 ft ²
D) 453.5 ft ²
E) 453.1 ft ²

Question 5

A rhombus has sides of length 100 centimeters, and the angle at one of the vertices is 70 . Approximate the lengths of the diagonals to the nearest tenth of a centimeter.

A) 125.5 cm , 164.6 cm
B) 114.7 cm , 163.8 cm
C) 124.9 cm , 164.6 cm
D) 124.9 cm , 163.6 cm
E) 34.7 cm , 34.7 cm

Question 6

A helicopter hovers at an altitude that is feet above a mountain peak of altitude 5,184 feet, as shown in the figure. A second, taller peak is viewed from both the mountaintop and the helicopter. From the helicopter, the angle of depression is , and from the mountaintop, the angle of elevation is . Approximate the altitude of the taller peak.

A) 5,437 ft
B) 5,439 ft
C) 5,441 ft
D) 4,931 ft
E) 5,443 ft

Question 7

The volume V of the right triangular prism shown in the figure is where B is the area of the base and h is the height of the prism. Approximate h, if .

A) 4.9
B) 18.8
C) 18.4
D) 18.0
E) 17.6

Question 8

Represent the complex number geometrically.

A)
B)
C)
D)
E)

Question 9

Determine whether the given pair of vectors are orthogonal or not. and

A) no
B) yes

Question 10

Find 4a + 6b. a = < 5, - 4>, b = <6, 5 >

A) 4a + 6b = < 9, 12 >
B) 4a + 6b = < 61, 19 >
C) 4a + 6b = < 56, 14 >
D) 4a + 6b = < 36, 30 >
E) 4a + 6b = < 51, 24 >

Question 11

A straight road makes an angle of with the horizontal. When the angle of elevation of the sun is , a vertical pole at the side of the road casts a shadow feet long directly down the road, as shown in the figure. Approximate the length of the pole.

A) 94.60 ft
B) 94.90 ft
C) 62.67 ft
D) 94.00 ft
E) 95.50 ft

Question 12

Solve

A)
B)
C)
D)
E) No triangle exists

Question 13

Find || a ||. a = - < 2, - 5 >

A) $\| a \| = \sqrt { 7 }$
B) $\| a \| = \sqrt { 29 }$
C) $\| a \| = \sqrt { 5 }$
D) $\| a \| = \sqrt { 28 }$
E) $\| a \| = \sqrt { 50 }$

Question 14

Two tugboats are towing a large ship into port, as shown in the figure. The larger tug exerts a force of 4,100 pounds on its cable, and the smaller tug exerts a force of 3,500 pounds on its cable. If the ship is to travel on a straight line l, approximate the angle that the larger tug must make with l.

A)
B)
C)
D)
E)

Question 15

Approximate the area of a parallelogram that has sides of lengths a and b (in feet) if one angle at a vertex has measure $\theta$ . $a = 9.0 , b = 16.0$ $\theta = 40 ^ { \circ }$

A) 95.2
B) 110.3
C) 89.3
D) 88.4
E) 92.6

Question 16

Given that , find the number.

A) -30
B) -60
C) 38
D) -42
E) 18

Question 17

Find (a) the dot product of the two vectors and (b) the angle between the two vectors. and

A)
B)
C)
D)
E)

Question 18

Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $5 i$ and $4 j$

A) $0,0 ^ { \circ } 0 ^ { \prime }$
B) $\text { 20, } 0 ^ { \circ } 0 ^ { \prime }$
C) $1,180 ^ { \circ } 0 ^ { \prime }$
D) $0,180 ^ { \circ } 0 ^ { \prime }$
E) $20,180 ^ { \circ } 0 ^ { \prime }$

Question 19

Solve

A)
B)
C)
D)
E) No triangle exists

Question 20

If c represents a constant force, find the work done if the point of application of c moves along the line segment from P to Q. and

A) 48
B) 12
C) -24
D) -12
E) 21

Question 21

Use De Moivre's theorem to change the given complex number to the form $a + b i$ , where $a$ and $b$ are real numbers. $( - 2 - 2 t ) ^ { 10 }$

A) $65,536$
B) $1,024 + 1,024 i$
C) $32,768 i$
D) $65,536 i$
E) $32,768$

Question 22

Find the four fourth roots of .

A)
B)
C)
D)
E)

Question 23

Use De Moivre's theorem to change the given complex number to the form , where and are real numbers.

A) 4,096
B) -24+24i
C) 2,048
D) -4,096i
E) -2,048i

Question 24

The trigonometric form of complex numbers is often used by electrical engineers to describe the current I, voltage V, and impedance Z in electrical circuits with alternating current. Impedance is the opposition to the flow of current in a circuit. Most common electrical devices operate on 115-volt, alternating current. The relationship among these three quantities is . The real part of I represents the actual current delivered to an electrical appliance in amps. Approximate this current when and , and express the answer in rectangular form to two decimal places.

A) 2.76 amps
B) 1.33 amps
C) 1.23 amps
D) 1.38 amps
E) 1.45 amps

Question 25

Use De Moivre's theorem to change the given complex number to the form , where and are real numbers.

A)
B)
C)
D)
E)