MHB "Bearing" question - finding distance between two objects

[tantrik]

Dear friends,

I am having problem solving the following question related to "bearing" in trigonometry.

A motorboat is 10 km South of a lighthouse and is on a course of 053 degrees. What is the shortest distance between the motorboat and the lighthouse?

My confusion is how does a boat travel 10 km South of a lighthouse and making an angle of 53 degrees clockwise from North. Is there something wrong with the question itself?

Thanks in advance.

 

[Ackbach]

I think the $10 \, \text{km}$ distance is the initial position of the motorboat. It'll travel in a direction $53^{\circ}$ clockwise of North, passing by the lighthouse. There will be one point in its travel where it will be the closest to the lighthouse. How far away is the lighthouse at that point?

 

[HallsofIvy]

Draw a line from the lighthouse "due south" to the present position of the ship. Draw the line showing the path of the ship. Finally, draw the line from the lighthouse perpendicular to the path of the ship. You now have a right triangle. The first line, "due south", is the hypotenuse and is 10km long. The two other lines are the legs of the right triangle. Since the path of the ship makes a 53 degree angle with north, the angle at the current position of the ship is 53 degrees.

So you have a right triangle with angle 53 degrees, the hypotenuse of length 10 and you want to find the length of the leg opposite that angle. Sounds like a sine!