Sine laws with Oblique Triangles: The Tower of Pisa

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SUMMARY

The discussion focuses on calculating the slant height and actual height of the Leaning Tower of Pisa using trigonometric principles. The tower leans at an angle of 5.5°, and its shadow measures 90 meters, with an angle of elevation of 32° from the shadow's tip to the tower's top. Participants clarify that the "slant height" refers to the length of the sloping side of the triangle formed by the tower's lean, while the actual height is a separate measurement. The sine law is utilized to derive these dimensions accurately.

PREREQUISITES
  • Understanding of trigonometric functions, specifically sine and cosine.
  • Familiarity with the sine law in triangle calculations.
  • Basic knowledge of oblique triangles and their properties.
  • Ability to interpret angles of elevation and their applications in real-world scenarios.
NEXT STEPS
  • Study the application of the sine law in oblique triangles.
  • Learn how to calculate angles of elevation and depression in practical problems.
  • Explore advanced trigonometric concepts related to non-right triangles.
  • Investigate real-world applications of trigonometry in architecture and engineering.
USEFUL FOR

Students studying trigonometry, educators teaching geometry, architects, and engineers involved in structural design and analysis.

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Here's the question: The leaning Tower of Pisa leans toward the south at an angle of 5.5°. On one day, its shadow was 90m long, and the angle of elevation from the tip of the shadow to the top of the tower is 32°.

Determine the slant height of the tower.

How high is the tip of the tower above the ground?

Now what I couldn't get is what is the "slant height?" I did the equation using the sine law, but then realized I had just found the length of the slant, not the height. Any suggestions?
 
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The "slant height" of a triangle is the height of one of its sloping sides, in this case the side where the leaning tower of pizza is leaning towards.

So, yes the length of the slant is the "slant height". The height of the leaning tower would be different (which was what you thought it was).
 

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