From the top of a 215-ft lighthouse, the angle of depression to a ship in the ocean is 76.6 degrees . How far is the ship from the base of the lighthouse? Round your answer to the hundredths place .

Answer 1

Hi Angel T.,

The angle of depression is the angle between the horizontal line of sight and the line of sight down to an object.

The horizontal line of sight from the top of the lighthouse and the vertical distance from the ship to this horizontal line of sight forms a right triangle.

The distance from the ship to the lighthouse is the same as the horizontal line of sight from a-top the lighthouse. This line of sight is the line adjacent (SA-side adjacent) to the angle of depression (θ).

And the distance from the bottom to the top of the lighthouse (215 ft.) is the same distance from the horizontal line of sight of the lighthouse straight down to the ship (SO-side opposite = 215).

We can write a trig. function to find the distance from the lighthouse to the ship:

tan(θ) = SO/SA, and we know θ=76.6^o, and we know SO=215, and we want to solve for SA:

SA = SO/tan(θ)

SA = 215/tan(76.6) = 51.22 ft.

I hope this helps, Joe.