4.8 Applications and Models 1

A ship leaves port at noon and heads due west at 20 knots (nautical miles per hour). At 2 pm, the ship changes course to N54˚W. Find the ship’s bearing and distance from the port of departure at 3 pm.

Draw and label a

sketch of the problem.

We will solve it together.

Warm-up

40nm

5420 nm

36x

y

2036sin

x

2036cos

y

75.11x

18.16y

d

= 2(20 nm)

(40 + 16.18)2 + (11.75)2 = d2

z

d = 57.39nm

SOLUTION: A ship leaves port at noon and heads due west at 20 knots (nautical miles per hour). At 2 pm the ship changes course to N54W. Find the ship’s bearing and distance from the port of departure at 3 pm

A ship leaves port at noon and heads due west at 20 knots (nautical miles per hour). At 2 pm the ship changes course to N54W. Find the ship’s bearing and distance from the port of departure at 3 pm

40nm

5420 nm

36x

y

2036sin

x

2036cos

y

75.11x

18.16y

z

4018.16

75.11tan

z

d

= 2(20 nm)

z = 12

or N 78 WW 12 N

Trig Game Plan Date: 10/8/13Trigonometry

Standard12.0 Students use trigonometry to determine unknown sides or angles in right triangles.

Section/Topic 2.5a Further Applications of Right Triangles

Objective Student will be able to under solve bearing problems using right triangles trigonometry

Homework P97, 23 – 28, 32 - 34

Announcements

Late Start, Wednesday 10/9/13Back to School, Wednesday 10/9/13Chapter 2 Test, Friday 10/11/13

From a given point on the ground, the angle of elevation to the top of a tree is 36.7˚. From a second point, 50 feet back, the angle of elevation to the top of the tree is 22.2˚. Find the height of the tree to the nearest foot.

Example 4 SOLVING A PROBLEM INVOLVING ANGLES OF ELEVATION

The figure shows two unknowns: x and h.

Since nothing is given about the length of the hypotenuse, of either triangle, use a ratio that does not involve the hypotenuse.

TANGENT

In triangle ABC:

Example 4 SOLVING A PROBLEM INVOLVING ANGLES OF ELEVATION (continued)

In triangle BCD:

Each expression equals h, so the expressions must be equal.

Example 4 SOLVING A PROBLEM INVOLVING ANGLES OF ELEVATION (continued)

Since h = x tan 36.7˚, we can substitute.

The tree is about 45 feet tall.

Example: Solving a Problem Involving Angles of Elevation

Sean wants to know the height of a Ferris wheel. From a given point on the ground, he finds the angle of elevation to the top of the Ferris wheel is 42.3˚. He then moves back 75 ft. From the second point, the angle of elevation to the top of the Ferris wheel is 25.4˚. Find the height of the Ferris wheel.

• The figure shows two unknowns: x and h.

• Since nothing is given about the length of the hypotenuse, of either triangle, use a ratio that does not involve the hypotenuse, (the tangent ratio).

• In triangle ABC,

• In triangle BCD,

xC

B

h

DA 75 ft

• Since each expression equals h, the expressions must be equal to each other.

Solve for x.

Distributive Property

Factor out x.

Get x-terms on one side.

Divide by the coefficient of x.

• We saw above that Substituting for x.

• tan 42.3 = .9099299 and tan 25.4 = .4748349. So, tan 42.3 - tan 25.4 = .9099299 - .4748349 = .435095 and

– The height of the Ferris wheel is approximately 75 ft.

Example: Solving a Problem Involving Angles of Elevation

• Jenn wants to know the height of a Ferris wheel. From a given point on the ground, she finds the angle of elevation to the top of the Ferris wheel is 49.6° . She then moves back 65 ft. From the second point, the angle of elevation to the top of the Ferris wheel is 29.7° . Find the height of the Ferris wheel.

• The figure shows two unknowns: x and h.

• Since nothing is given about the length of the hypotenuse, of either triangle, use a ratio that does not involve the hypotenuse, (the tangent ratio).

• In triangle ABC,

• In triangle BCD,

xC

B

h

DA 75 ft

x

• Since each expression equals h, the expressions must be equal to each other.

Solve for x.

Distributive Property

Factor out x.

Get x-terms on one side.

Divide by the coefficient of x.

xC

B

h

DA 75 ft

x

x≈61.32 ft

Marla needs to find the height of a building. From a given point on the ground, she finds that the angle of elevation to the top of the building is 74.2°. She then walks back 35 feet. From the second point, the angle of elevation to the top of the building is 51.8°. Find the height of the building.

• There are two unknowns, the distance from the base of the building, x, and the height of the building, h.

Solving a Problem Involving Angles of Elevation (cont.)

In triangle ABC

In triangle BCD

• Set the two expressions for h equal and solve for x.

Since h = x tan 74.2°, substitute the expression for x to find h.

The building is about 69 feet tall.