How Far is the Ship From Harbor?

[Guest]

The sailor of a ship at sea has the 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 to be 20 degrees. With the ship heading directly toward the harbor, she repeats this measurement after 5 minutes of traveling at 12 miles per hour.

If the new angle is 30 degrees, how far is the ship from the harbor?

 

[tkhunny]

You can do it.

The "1 mile" from lighthouse to harbor doesn't change.

Distance to the harbor changes by [5 min * 12 mph * (1 hr / 60 min)] = 1 mile

Put on your drawing hat and draw the picture.
Put on your trig hat and work it out.

 

[soroban]

Hello, interval!

I'll help with the drawings . . .

The sailor of a ship at sea has the 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 to be 20°.

With the ship 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?

      H             1 mile                L
      o - - - - - - - - - - - - - - - - - o
      :                               *
      :                           *
      :                       *
    x :                   *
      :               *
      :           *
      : 20°   *
      :   *
    A o

The ship is at \(\displaystyle A,\;x\) miles from the harbor \(\displaystyle (H).\)
The lighthouse is at \(\displaystyle L:\;HL \,=\, 1\) mile . . . and\(\displaystyle \angle\,A\,=\,20^o.\)

      H             1 mile                L
      o - - - - - - - - - - - - - - - - - o
      :                             *
      :                       *
  x-1 :                 *
      : 30°       *
      :     *
    B o
      :
      :1
      :
    A *

After 5 minutes at 12 mph, the ship is at \(\displaystyle B\), 1 mile closer to \(\displaystyle H.\)
Hence: \(\displaystyle \,BH \:=\:x\,-\,1\) . . . and\(\displaystyle \angle\,B\,=\,30^o.\)


Can you finish the problem now?