A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed, the angle of depression to the boat is 17°44'. When the boat stops, the angle of depression is 48°13'. The lighthouse is 200 feet tall. How far did the boat travel from when it was first noticed until it stopped? Round your answer to the hundredths place.

Question

Mathematics

Kayla Swanson

14 October, 06:07

A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed, the angle of depression to the boat is 17°44'. When the boat stops, the angle of depression is 48°13'. The lighthouse is 200 feet tall. How far did the boat travel from when it was first noticed until it stopped? Round your answer to the hundredths place.

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Answers (1)

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Phoenix Foster · Answer 1

446.71 feet

Step-by-step explanation:

17°44' means 17 degrees and 44 minutes. A minute is 1/60 of a degree.

17°44' = 17 + (44/60) = 17.733°

Similarly:

48°13' = 48 + (13/60) = 48.217°

When the boat is first noticed:

tan (17.733°) = 200 / a

a = 200 / tan (17.733°)

a = 625.43

When the boat stops:

tan (48.217°) = 200 / b

b = 200 / tan (48.217°)

b = 178.72

So the difference is:

a - b = 625.43 - 178.72 = 446.71

The boat traveled 446.71 feet from the time it was first noticed to the time it stopped.