How Far is the Ship from the Rocks When Sighting a Lighthouse?

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SUMMARY

The problem involves calculating the distance from a ship to a rocky cliff based on the height of a lighthouse and the angle of sighting. The lighthouse stands 49 feet tall, with a cliff extending 19 feet horizontally from its base. A sailor's eye level is 14 feet above the water, creating a right triangle where the opposite side measures 35 feet. The correct distance from the ship to the rocks is determined using the tangent function, resulting in an approximate distance of 60.6 feet after adjusting for the cliff's horizontal extension.

PREREQUISITES
  • Understanding of trigonometric functions, specifically tangent.
  • Knowledge of right triangle properties.
  • Ability to interpret word problems in geometry.
  • Familiarity with basic algebra for solving equations.
NEXT STEPS
  • Review trigonometric identities and their applications in real-world problems.
  • Practice solving right triangle problems using the tangent function.
  • Explore geometric interpretations of word problems to enhance comprehension.
  • Learn about the implications of height and distance in navigation and maritime contexts.
USEFUL FOR

Students studying geometry, sailors or navigators needing to calculate distances, and educators teaching trigonometry and its applications in real-life scenarios.

cphill29
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Homework Statement



A lighthouse that rises 49 ft above the surface of the water sits on a rocky cliff that extends 19 feet from its base. A sailor on the deck of a ship sights the top of the lighthouse at an angle of 30 degrees above the horizontal. If the sailor's eye level is 14 ft above the surface of the water, how far is the ship from the rocks?

Homework Equations



tanx=opp/adj

The Attempt at a Solution



tan30=35/x
xtan30=35
x=35/(tan30)
x=60.6 ft

Since the lighthouse is 49 feet above the surface and the sailor is 14 feet above the surface, the opposite side of the triangle is 35 feet, but I can't get the right answer doing this.
 
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The wording of the problem is not that clear. I believe when saying that ' the rock cliff extends 19 feet from its base ' , that it means that the rocks start 19 feet horizontally away the lighthouse. So your method is correct, just make the necessary adjustment to your answer.
 
PhanthomJay said:
The wording of the problem is not that clear. I believe when saying that ' the rock cliff extends 19 feet from its base ' , that it means that the rocks start 19 feet horizontally away the lighthouse. So your method is correct, just make the necessary adjustment to your answer.

That's what I thought, too, but it gives me a picture and it's clear that is not the case. I did try it though and still got the answer wrong.
 

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