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 54  20 nm 36  x y d = 2(20 nm) ( ) 2 + (11.75) 2 = d 2 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 N54  W. 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 N54  W. Find the ship’s bearing and distance from the port of departure at 3 pm 40nm 54  20 nm 36  x y z d = 2(20 nm) z = 12  or N 78  W W 12  N

Trig Game Plan Date: 10/8/13 Trigonometry Standard 12.0 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, Announcements Late Start, Wednesday 10/9/13 Back to School, Wednesday 10/9/13 Chapter 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.

Graphing Calculator Solution Superimpose coordinate axes on the figure with D at the origin. The coordinates of A are (50, 0). The tangent of the angle between the x-axis and the graph of a line with equation y = mx + b is the slope of the line. For line DB, m = tan 22.2°. Example 4 SOLVING A PROBLEM INVOLVING ANGLES OF ELEVATION (continued) Since b = 0, the equation of line DB is

Use the coordinates of A and the point-slope form to find the equation of AB: Example 4 SOLVING A PROBLEM INVOLVING ANGLES OF ELEVATION (continued) The equation of line AB is

Graph y 1 and y 2, then find the point of intersection. The y-coordinate gives the height, h. The building is about 45 feet tall. Example 4 SOLVING A PROBLEM INVOLVING ANGLES OF ELEVATION (continued)

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, x C 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 = and tan 25.4 = So, tan tan 25.4 = = 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, x C 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.

x C 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.