From the Top of a 7m High Building

Analyzing Height Calculation Patterns

Before diving into the main topic, let's explore some common patterns in height determination:

  • Finding Heights: Given an elevation angle and a point of observation, the tangent ratio helps determine a tower's height. The tower's height can be adjusted through queries.
  • Tower Height with Flagstaff: When the flagstaff's height is known, the tower's peak can be found. The distance between the observation point and the base remains constant. Calculations can involve height or distance from the base.
  • Wind-blown Tree: This pattern involves finding the height of a tree broken by wind, forming a right-angled triangle. Calculations can involve the tree's original height or the distance from the broken part's highest point.
  • Elevation Angle: Tangent ratios determine the elevation angle when the tower’s height and observation point are available.
  • Shadow Length: The tower/pole's shadow length is crucial. As elevation angles decrease, shadow size increases. Calculations involve shadow length or object height.
  • Cloud Height: Trigonometric ratios, using the depression angle, calculate the height of a cloud above a lake. Questions involve the cloud's depression angle and height above the lake.
  • Tower Height with Two Observation Points: When two elevation angles exist, the constant tower height aids in calculating the building's height by asking questions from various angles.

Calculating Tower Height from a 7m Building

Consider this example: From a 7-meter-high building, the angle of elevation to a tower is 60°, and the angle of depression to its base is 45°.

Here's the information we have:

  • Building Height: 7m
  • Angle of Elevation: 60°
  • Angle of Depression: 45°

Calculations:

  1. tan(45°) = AB/BC
  2. 1 = AB/BC
  3. AB = BC = 7m
  4. tan(60°) = ED/AD
  5. √3 = ED/7 (Since AD = BC)
  6. ED = 7√3 m
  7. Tower Height: ED + CD
  8. Tower Height: 7√3 + 7 (Since AB = CD)
  9. Tower Height: Approximately 19.12m

Therefore, the tower’s height is approximately 19.12 meters.