Example 3

9.1.3 Example 3

Description

Quick Overview

This section illustrates how to calculate the height of a chimney using trigonometry based on the observer's position and angle of elevation.

Standard

In this example, an observer measures the angle of elevation to the top of a chimney from a distance. Using trigonometric functions, specifically the tangent of the angle, we determine the height of the chimney based on the observer's height and distance from the chimney.

Detailed

Example 3: Calculating the Height of a Chimney Using Trigonometry

In this example, we explore how to determine the height of a chimney based on an observer's distance from it and the angle of elevation to its top. The observer is 1.5 meters tall and positioned 28.5 meters away from the chimney. The angle of elevation, measured from the observer's line of sight, is 45 degrees.

To solve this problem, we recognize that the scenario can be represented as a right triangle where:
- AB (the chimney's height) can be broken down into two parts: the height above the observer's eyes (AE) and the observer's height (BE).
- The horizontal distance from the observer to the base of the chimney is represented by DE.

Using the tangent ratio, we have the relationship: tan(angle) = opposite/adjacent. Here, tan(45°) = AE / DE. Since tan(45°) = 1, we find that AE = DE = 28.5 m.
Combining this with the height of the observer, the total height of the chimney (AB) is calculated as: AB = AE + BE = 28.5 m + 1.5 m = 30 m.
Thus, the height of the chimney is 30 meters. This example highlights the practical application of trigonometric ratios in solving real-world measurement problems.

Key Concepts

  • Height Calculation: The process of determining the height of an object using the angle of elevation and distance.

  • Trigonometric Ratios: Functions like tangent that relate angle measures to side lengths in right triangles.

  • Right Triangle Properties: Understanding the relationship between the sides and angles in right triangles.

Memory Aids

🎵 Rhymes Time

  • To find the height of a tower or tree, just look up high and let angles be free.

📖 Fascinating Stories

  • Once there was a curious observer who hailed from a hill. She measured a chimney tall and got quite a thrill. With angles and distance and her height in the way, she calculated the total height without any delay.

🧠 Other Memory Gems

  • A mnemonic to remember: AE is added to BE for total height, A + B = all right.

🎯 Super Acronyms

Use H = A + B for height calculation

  • H: for height
  • A: for above eyes
  • B: for below.

Examples

  • An observer standing 1.2 meters tall wants to find the height of a tree 35 meters away with an angle of elevation of 60 degrees. Using the tangent function, we can determine the height.

  • If a building is observed from a distance of 50 meters with an angle of elevation of 30 degrees, what would be its height using trigonometric ratios?

Glossary of Terms

  • Term: Angle of Elevation

    Definition:

    The angle between the horizontal and the line of sight when looking up at an object.

  • Term: Tangent

    Definition:

    A trigonometric ratio used to relate the angle of elevation to the opposite side and adjacent side of a right triangle.

  • Term: Right Triangle

    Definition:

    A triangle that has one angle measuring 90 degrees.

  • Term: Observer

    Definition:

    A person or entity viewing an object from a specified distance and height.