Question

A boat is heading towards a lighthouse, where Dominic is watching from a vertical distance of 114 feet above the water. Dominic measures an angle of depression to the boat at point AA to be 6degrees ∘ . At some later time, Dominic takes another measurement and finds the angle of depression to the boat (now at point BB) to be 32degrees ∘ . Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.

Answer 1

902 ft

Explanation:

You want the distance from point A to point B if the angles of depression from a point 114 ft above their level are 6° and 32°, respectively.

Tangent

The tangent function is related to triangle sides by ...

Tan = Opposite/Adjacent

Referring to the attached, we can see that the angles of depression are the complement of the angles adjacent to the height of the observer. This means we have ...

tan(90° -6°) = AO/OD = AO/114 ⇒ AO = 114·tan(84°)

tan(90° -32°) = BO/OD = BO/114 ⇒ BO = 114·tan(58°)

Distance

The distance between points A and B is ...

AB = AO -BO = 114(tan(84°) -tan(58°)) ≈ 902 . . . . feet

The distance from point A to point B is about 902 feet.

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