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**Right Triangle Problems**

Sec. 4.8

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**Angle of Depression Angle of Elevation**

New Definitions Angle of Elevation – angle through which the eye moves up from horizontal to look at something above. Angle of Depression – angle through which the eye moves down from horizontal to look at something below. Consider two people looking at each other from the top and bottom of a flight of stairs: Angle of Depression Angle of Elevation

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Guided Practice The angle of depression of a buoy from the top of a lighthouse 130 feet above the surface of the water is Find the distance x from the base of the lighthouse to the buoy. First, draw a diagram: Solve for x algebraically: 6 130 ft x 6 feet

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Guided Practice From the top of a 100-ft tall building a man observes a car moving toward the building. If the angle of depression of the car changes from 22 to 46 during the period of observation, how far does the car travel? The diagram: 22 46 100 ft 46 22 x d From the smaller right triangle:

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Guided Practice From the top of a 100-ft tall building a man observes a car moving toward the building. If the angle of depression of the car changes from 22 to 46 during the period of observation, how far does the car travel? The diagram: 22 46 100 ft 46 22 x d Substitute!!! From the larger right triangle:

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**Guided Practice x d feet**

From the top of a 100-ft tall building a man observes a car moving toward the building. If the angle of depression of the car changes from 22 to 46 during the period of observation, how far does the car travel? The diagram: 22 46 100 ft 46 22 x d feet

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Guided Practice A large, helium-filled penguin is moored at the beginning of a parade route. Two cables attached to the underside of the penguin make angles of 48 and 40 with the ground and are in the same plane as a perpendicular line from the penguin to the ground. If the cables are attached to the ground 10 feet from each other, how high above the ground is the penguin? Definition of the tangent function: The diagram: Solve both for h: h 40 48 Set equal, solve for x: 10 x

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Guided Practice A large, helium-filled penguin is moored at the beginning of a parade route. Two cables attached to the underside of the penguin make angles of 48 and 40 with the ground and are in the same plane as a perpendicular line from the penguin to the ground. If the cables are attached to the ground 10 feet from each other, how high above the ground is the penguin? The diagram: h 40 48 10 x Plug back in to solve for h!!!

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**Guided Practice h feet 10 x**

A large, helium-filled penguin is moored at the beginning of a parade route. Two cables attached to the underside of the penguin make angles of 48 and 40 with the ground and are in the same plane as a perpendicular line from the penguin to the ground. If the cables are attached to the ground 10 feet from each other, how high above the ground is the penguin? The diagram: h 40 48 feet 10 x

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Guided Practice The top row of the red seats behind home plate at Cincinnati’s Riverfront Stadium is 90ft above the level of the playing field. The angle of depression to the base of the left field wall is 14 . How far is the base of the left field wall from a point on level ground directly below the top row? d = distance in question feet

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Guided Practice The angle of elevation of the top of Proctor’s ego is measured to be at a point 12.3 feet from the base of the ego. How tall is Proctor’s ego? h = height of Proctor’s ego feet

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Guided Practice While hiking on a level path toward Colorado’s front range, Otis Evans determines that the angle of elevation to the top of Long’s Peak is Moving 1000 ft closer to the mountain, Otis determines the angle of elevation to be How much higher is the top of Long’s Peak than Otis’s elevation? h = height in question feet

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**70 105 A U.S. Coast Guard patrol boat leaves Port Cleveland and**

averages 35 knots (nautical mph) traveling for 2 hours on a course of 53 and then 3 hours on a course of What is the boat’s bearing and distance from Port Cleveland? First, let’s diagram the path of the boat… Now, we need the measure of angle ABC… B 143 How about the distances AB and BC? 70 Distance = Speed x Time 53 53 37 105 AB = (35 knots)(2 hours) = 70 naut. mi. A BC = (35 knots)(3 hours) = 105 naut. mi. C

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**70 105 A U.S. Coast Guard patrol boat leaves Port Cleveland and**

averages 35 knots (nautical mph) traveling for 2 hours on a course of 53 and then 3 hours on a course of What is the boat’s bearing and distance from Port Cleveland? Solve the right triangle for AC and : B 143 70 naut. mi. 53 53 37 105 A C

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**The patrol boat is about 126 nautical miles from port, at a **

A U.S. Coast Guard patrol boat leaves Port Cleveland and averages 35 knots (nautical mph) traveling for 2 hours on a course of 53 and then 3 hours on a course of What is the boat’s bearing and distance from Port Cleveland? Interpret our answer: B 143 The patrol boat is about 126 nautical miles from port, at a bearing of approximately 70 53 53 37 105 A C

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**More Practice Problems**

A boat travels at 40 knots from its home port on a course of 65 for 2 hours and then changes to a course of for 4 hours. Find the distance and bearing from the port to the boat. 80 nautical miles 160 Port Bearing

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**More Practice Problems**

A ranger spots a fire from a 73-ft tower in Yellowstone National Park. She measures the angle of depression to be How far is the fire from the tower? 73 ft

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**More Practice Problems**

A six meter ladder makes an angle of 70 with the ground as it leans against a wall. The ladder then slips slightly so that the angle it makes with the ground changes to How much higher on the wall was the ladder before it slipped? Before slipping: After slipping: 6 m 6 m

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**More Practice Problems**

A six meter ladder makes an angle of 70 with the ground as it leans against a wall. The ladder then slips slightly so that the angle it makes with the ground changes to How much higher on the wall was the ladder before it slipped? Total change in vertical position on wall: meters

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