Indirect Measurement- Angles of depression or elevation can be used to determine speed

Question

a sailor wants to know how fast her sailboat is sailing. to do so she must determine how far she has moved in 15 minutes. the sailboat is moving toward the lighthouse that is 197 feet tall. from her first location, she knows that her angle of elevation to the top of the lighthouse is 1 degree. from her next location 15 minutes later, the angle of elevation to the top of the lighthouse is 5 degrees.

1. How far away from the lighthouse is the sailor at her first location?
2. How far away from the lighthouse is the sailor at her second location?
3. how far did the boat travel in 15 minutes?
4. what is the boats speed? round your answer to the nearest tenth of a foot per minute.
5. how will it take the boat to reach the lighthouse if the speed stays constant and the boat moves in the same direction?
6. If the sailor discovers the lighthouse is actually 175 ft tall, how fast is the boat moving?

Answer 1

1. tan(1 degree) = 197/x. x = 11,286.12 feet
2. tan(5 degrees) = 197/y. y= 2,251.72 feet
3. x-y = 9,034.4 feet
4. 9,034.4 feet/15 minutes = 602.3 feet/minute
5. 2,251.72 feet/ 602.3 feet per minute =3.74 minutes, approximately 3 minutes and 44 seconds
6. same thing over using 175 versus 197 gets you (10025.74-2000.26)/15 = 535 feet/minute