Question

The angle of depression from the top of a lighthouse to a boat in the ocean is 44°. If the lighthouse

is 245 feet tall, how far is the boat from the base of the lighthouse? Round to the nearest tenth.

Answer 1

The angle of depression is always measured from the horizontal.

A triangle is formed by the lighthouse, the ground and the boat. The angle of depression is not in the triangle, but it is equal to the angle of elevation at the boat. (Alternate angles on parallel lines)

The angle at the top of the lighthouse, inside the triangle is:

`90^\circ-27^\circ=63^\circ`

The distance to the boat is the side opposite the angle of 63° while the height of the lighthouse is the adjacent side.

`\frac{\text{opposite}}{\text{adjacent}} =\text{tan 63}^\circ`

`\frac{\text{distance}}{245} =\text{tan 63}^\circ`

`\text{distance}=245* \text{tan 63}^\circ`

`=480.83=480.84\thickapprox\boxed{\bold{480.8}}`