Question

A boat heading out to sea starts out at Point A, at a horizontal distance of 1032 feet

from a lighthouse/the shore. From that point, the boat's crew measures the angle of
elevation to the lighthouse's beacon-light from that point to be 15°. At some later
time, the crew measures the angle of elevation from point B to be 6°. Find the
distance from point A to point B. Round your answer to the nearest tenth of a foot if
necessary.

Answer 1

1598.9 ft

Explanation:

You want the distance between points A and B when observers at each point find the angle of elevation to a lighthouse beacon to be 15° and 6°, respectively. Point A is 1032 ft horizontal distance from the lighthouse.

Tangent

The tangent relation for sides in a right triangle is ...

Tan = Opposite/Adjacent

This can be rearranged to ...

Opposite = Tan × Adjacent

Setup

If d is the distance in feet from A to B, two expressions for the same lighthouse height can be written:

height = tan(15°)(1032)

height = tan(6°)(1032 +d)

Solution

Equating these values of height, we have ...

tan(6°)(1032 +d) = tan(15°)(1032)

d = 1032(tan(15°) -tan(6°))/tan(6°) = 1032(tan(15°)/tan(6°) -1)

d ≈ 1598.946 . . . . feet

The distance from A to B is about 1598.9 feet.