SPRECALC7 6.2.056.MI.

Question

From the top of a 260-ft lighthouse, the angle of depression to a ship in the ocean is 21°. How far is the ship from the base of the lighthouse? (Round your answer to the nearest foot.)

Answer 1

Hi Ashton!

The best thing to do is set up a diagram to better understand this question. Since this response is in text format and not a video, I will provide a link to a picture and try to explain further:

Angle of Depression - Definition, Formula, Examples (cuemath.com)

(Go to "Angle of Depression in a Triangle")

Now let's say the lighthouse is the AB line and the angle of depression is a. We also know that we are looking for the distance from the ship in the water the base of the lighthouse, so we are essentially looking for the length of BC.

We know the height of the lighthouse is 260 feet and the angle of depression is 21 degrees.

We are assuming the angle ∠ABC is 90 degrees (a right angle), giving us a right triangle. The reason why ∠ACB is also a (21 degrees) is because the lines BC and AD are parallel (by using trigonometric rules). So now we need to find the length of BC, which can be found using trigonometric functions, specifically tangent (tan). Tangent of an angle is equal to the length of the line opposite the angle divided by the length of the line adjacent to the angle (not the hypotenuse). Thus, we can set up our function as follows:

tan(a) = AB/BC

tan(21°) = 260/BC

Using a graphing calculator, we know tan(21°) = 0.38386403... Now we need to solve for BC. To get to the below step, we are multiplying both sides by BC and dividing both sides by tan(21°).

BC = 260/tan(21°) = 260/0.38386403 = 677.32

Therefore, the ship is 677.32 feet away from the base of the lighthouse.

Hope this helps!

Owen D.