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### 4.8 Trigonometric Applications and Models 2014

### 4.8 Trigonometric Applications and Models Day 2

Objectives:

Use right triangles to solve real-life problems.

Use directional bearings to solve real-life problems.

Use harmonic motion to solve real-life problems.

Observer

Angle of depression

Object

Object

Angle of elevation

Observer

Horizontal

Terminology- Angle of elevation – angle from the horizontal upward to an object.

- Angle of depression – angle from the horizontal downward to an object.

4.8 Applications and Models

Example – Solving Rt. Triangles

At a point 200 feet from the base of a building,

the angle of elevation to the bottom of a smokestack

is 35°, and the angle of elevation to the top of the

smokestack is 53°. Find the height of the smokestack.

4.8 Applications and Models

You try:

- A swimming pool is 20 meters long and 12 meters wide. The bottom of the pool is slanted so that the water depth is 1.3 meters at the shallow end and 4 meters at the deep end, as shown. Find the angle of depression of the bottom of the pool.

7.69°

W

E

70°

S

Trigonometry and Bearings- In surveying and navigation, directions are generally given in terms of bearings. A bearing measures the acute angle that a path or line of sight makes with a fixed north-south line.

N

35°

W

E

S

Example – Finding Directions Using Bearings

- A hiker travels at 4 miles per hour at a heading of S 35° E from a ranger station. After 3 hours how far south and how far east is the hiker from the station?

4.8 Applications and Models

Example – Finding Directions Using Bearings

A ship leaves port at noon and heads due west at 20 knots, or 20 nautical miles (nm) per hour. At 2 P.M. the ship changes course to N 54o W. Find the ship’s bearing and distance from the port of departure at 3 P.M.

d

20sin(36o)

a

54o

20 nmph for 1 hr

78.181o

20 nm

36o

θ

b

20cos(36o)

40 nm

20 nmph for 2 hrs

Bearing: N 78.181o W

A ship leaves port at noon and heads due west at 20 knots, or 20 nautical miles (nm) per hour. At 2 P.M. the ship changes course to N 54o W. Find the ship’s bearing and distance from the port of departure at 3 P.M.

d

a

54o

20 nm

78.181o

20sin(36o)

20 nmph for 1 hr

36o

θ

b

20cos(36o)

40 nm

20 nmph for 2 hrs

Bearing: N 78.181o W

Two lookout towers are 50 kilometers apart. Tower A is due west of tower B. A roadway connects the two towers. A dinosaur is spotted from each of the towers. The bearing of the dinosaur from A is N 43o E. The bearing of the dinosaur from tower B is N 58o W. Find the distance of the dinosaur to the roadway that connects the two towers.

h

43o

58o

47o

32o

A

B

x

50– x

Two lookout towers spot a fire at the same time. Tower B is Northeast of Tower A. The bearing of the fire from tower A is N 33o E and is calculated to be 45 km from the tower. The bearing of the fire from tower B is N 63o W and is calculated to be 72 km from the tower. Find the distance between the two towers and the bearing from tower A to tower B.

a

c

63o

72

d

B

b

45

s

33o

45cos(330) – 72sin(630)

b – d

A

a + c

45sin(330) + 72sin(630)

Homework

4.8 p 326

1, 5, 9, 17-37 Odd

Quiz tomorrow on sections 4.5,4.6, and 4.7

4.8 Applications and Models

Objectives:

Use harmonic motion to solve real-life problems.

HWQ 11/14

- A plane is 160 miles north and 85 miles east of an airport. The pilot wants to fly directly to the airport. What bearing should be taken?

4.8 Applications and Models

Terminology

- Harmonic Motion – Simple vibration, oscillation, rotation, or wave motion. It can be described using the sine and cosine functions.
- Displacement – Distance from equilibrium.

4.8 Applications and Models

Simple Harmonic Motion

- A point that moves on a coordinate line is in simple harmonic motion if its distance d from the origin at time t is given by

where a and ω are real numbers (ω>0)

and frequency is number of cycles per unit of time.

4.8 Applications and Models

Example – Simple Harmonic Motion

Given this equation for simple harmonic motion

Find:

- Maximum displacement
- Frequency
- Value of d at t=4
- The least positive value of t when d=0

4.8 Applications and Models

You Try – Simple Harmonic Motion

- A mass attached to a spring vibrates up and down in simple harmonic motion according to the equation
- Find:
- Maximum displacement
- Frequency
- Value of d at
- 2 lvalues of t for which d=0

Example – Simple Harmonic Motion

- A weight attached to the end of a spring is pulled down 5 cm below its equilibrium point and released. It takes 4 seconds to complete one cycle of moving from 5 cm below the equilibrium point to 5 cm above the equilibrium point and then returning to its low point.
- Find the sinusoidal function that best represents the motion of the moving weight.
- Find the position of the weight 9 seconds after it is released.

You Try – Simple Harmonic Motion

- A buoy oscillates in simple harmonic motion as waves go past. At a given time it is noted that the buoy moves a total of 6 feet from its low point to its high point, returning to its high point every 15 seconds.
- Write a sinusoidal function that describes the motion of the buoy if it is at the high point at t=0.
- Find the position of the buoy 10 seconds after it is released.

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