Distances at Sea

greatwhiteshark

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May 8, 2005
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The navigator of a ship at sea has a harbor in sight at which the ship is to dock. She spots a lighthouse that she knows is 1 mile up the coast from the mouth of the harbor and she measures the angle between the line-of-sight observations of the harbor and lighthouse to be 20 degrees. With the ship heading directly toward the harbor, she repeast this measurement after 5 minutes of traveling at 12 mph. If the new angle is 30 degrees, how far is the ship from the harbor?
 
Hello, greatwhiteshark!

This is similar to other problems . . .

The navigator of a ship at sea has a harbor in sight at which the ship is to dock.
She spots a lighthouse that she knows is 1 mile up the coast from the mouth of the harbor
and she measures the angle between the lines-of-sight of the harbor and lighthouse to be 20°.
Heading directly toward the harbor, she repeats this measurement after 5 minutes of traveling at 12 mph.
If the new angle is 30°, how far is the ship from the harbor?
Code:
    H             1               L  
    +-----------------------------*     The ship starts at A.                       
    |                      *   *        The harbor is H.
  x |                *     *            The lighthouse is L.
    |          *       *                Angle HAL is 20°
    |30° *         *            
  B *           *                       Later the ship is at B.
    |        *                          Angle HBL = 30°
  1 |20°  *                         
    |  *                    
  A *
Moving at 12 mph, in 5 minutes (1/12 hour), it travels 1 mile.
. . . Hence: AB = 1

There are two right triangles.
Use the tangent function to solve for x.
 
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