Question

A boat is heading towards a lighthouse, whose beacon-light is 105 feet above the water. From point AA, the boat’s crew measures the angle of elevation to the beacon, 7^{\circ}

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, before they draw closer. They measure the angle of elevation a second time from point BB at some later time to be 21^{\circ}
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. Find the distance from point AA to point BB. Round your answer to the nearest tenth of a foot if necessary.

Answer 1

AB ≈ 581.6 ft

Explanation:

The triangle from point A to the base of the lighthouse to the light beam can be modeled as a right triangle with the side opposite the angle being given. That means the distance from A to the base of the lighthouse is found from the tangent relation:

Tan = Opposite/Adjacent

tan(7°) = (105 ft)/A

A = (105 ft)/tan(7°)

Using similar reasoning, the distance to B is ...

B = (105 ft)/tan(21°)

Then the distance from A to B is ...

AB = (105 ft)/tan(7°) -(105 ft)/(tan(21°)) = (105 ft)(1/tan(7°) -1/tan(21°))

AB ≈ 581.6 ft

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Additional comment

We know that 1/tan(7°) = cot(7°) = tan(83°). This means the relation can be written ...

AB = (105 ft)(tan(83°) -tan(69°))