Example 7

9.1.7 Example 7

Description

Quick Overview

This section involves calculating the width of a river using angles of depression from a bridge.

Standard

In Example 7, students learn how to determine the width of a river based on angles of depression observed from a bridge. The example illustrates the application of trigonometric ratios in right triangles, specifically using angles of depression and the height of the bridge.

Detailed

Example 7 Analysis

In this example, a person stands on a bridge that is 3 meters high and records the angles of depression to the banks of a river on both sides. The angles of depression are 30° and 45°, respectively. By applying the principles of right triangle trigonometry, we can determine the width of the river by calculating the horizontal distances to each bank from the foot of the bridge.

Definitions and Trigonometric Relationships Used:
- Angle of Depression: The angle formed by a horizontal line and a line of sight down to an object below the line.
- Tan Function: Utilized in the right triangles formed by the height of the bridge and the distances to the banks.

In the analysis:
1. Triangle APD uses angle 30° to calculate distance AD.
2. Triangle PBD uses angle 45° to find distance BD.

Finally, the total width of the river AB is the sum of AD and BD, yielding the final result in the form of 3(1 + √3)m.

Key Concepts

  • Angle of Depression: The angle formed with a horizontal line when looking down to an object below.

  • Height of the Bridge: The vertical distance from the bridge to the ground or water level.

  • Tan Function: A ratio used in trigonometry to relate angles and lengths of triangles.

Memory Aids

🎵 Rhymes Time

  • If you see the bank, look down the plank; at thirty degrees you’ll see the trees!

📖 Fascinating Stories

  • Imagine a bridge where a person looks down, seeing a beautiful river flowing below. The angles guide them to calculate the distance with ease, like a treasure hunt under the trees.

🧠 Other Memory Gems

  • When calculating lengths, remember: TAD (Tangent for Angle and Distance).

🎯 Super Acronyms

Use `TAD` to remember Tangent, Angle, Distance – the key to solving!

Examples

  • Consider a building that is 10 meters tall. If the angle of depression to the ground from the top of the building is 60°, what is the distance from the base of the building to the point directly below the top of the building?

  • From a hill 20 m high, if the angle of depression to a car is 30°, calculate the distance from the base of the hill to the car.

Glossary of Terms

  • Term: Angle of Depression

    Definition:

    The angle formed by a horizontal line and a line of sight from above to a point below the horizontal.

  • Term: Height

    Definition:

    The measure of how tall something is from its base to its top, here referring to the height of the bridge.

  • Term: Tan Function

    Definition:

    A trigonometric function that relates the angle of a right triangle to the ratio of the opposite side to the adjacent side.